tag:blogger.com,1999:blog-10411796536688729952024-03-13T22:49:35.387-07:00MATEMATICA FUNCIONES: GRÁFICA, DOMINIO , RANGO, FUNCION POLINOMICA, RAIZ INDICE PAR IMPAR RACIONALTeoría, ejercicios sobre funciones, dominio, rango y gráfica de funciones
Funciones polinomica, raíz de indice par, indice impar y racional
Matemática Basica SuperiorVídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-1041179653668872995.post-37983274813764972902019-01-08T12:23:00.000-08:002019-01-08T12:23:06.364-08:00FUNCIONES CUADRATICAS, GRAFICA, DOMINIO Y RANGO<b style="font-family: arial, helvetica, sans-serif;"><span style="color: red;">Blog de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/ox9aJg"><span style="color: blue;">https://goo.gl/ox9aJg</span></a></b><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Vídeos de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/PtnBkt"><span style="color: blue;">https://goo.gl/PtnBkt</span></a><span style="color: red;"><o:p></o:p></span></b></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Blog de matemática QuidiMat, teoría, ejemplos y problemas: </span><span style="color: blue;"><a href="https://goo.gl/wkR8Rc">https://goo.gl/wkR8Rc</a></span><span style="color: red;"><o:p></o:p></span></b></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b>Funciones cuadráticas, gráfica, dominio y rango: <a href="https://youtu.be/Mt2_eebcRWY">https://youtu.be/Mt2_eebcRWY</a></b></span><br />
<b style="text-indent: -22.65pt;"><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;">a)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; font-weight: normal; line-height: normal;"> </span></span></b><b style="text-indent: -22.65pt;"><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;">FUNCIÓN CUADRÁTICA</span></b><br />
<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 21.25pt;"> A la función f, le llamaremos función cuadrática, si su regla de correspondencia es:</span><br />
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">f(x)= ax<sup>2</sup>+bx+c donde x </span><span style="font-family: "symbol"; font-size: 11pt;">Î</span><span style="font-family: "arial" , sans-serif; font-size: 11pt;"> R.</span></div>
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<b><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt; text-indent: -18pt;">A)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt; text-indent: -18pt;">FORMA CANÓNICA:</span></b></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">f(x) = x<sup>2</sup> coordenadas del vértice: V(0;0)<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;"><b>Ejemplo:<o:p></o:p></b></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Bosqueja su gráfica, calcula dominio y rango de f(x) = x<sup>2</sup><o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Dom(f) = R Rang(f) = </span><span style="font-family: "symbol"; font-size: 11pt;">[</span><span style="font-family: "arial" , sans-serif; font-size: 11pt;">0; +</span><span style="font-family: "symbol"; font-size: 11pt;">¥</span><span style="font-family: "symbol"; font-size: 11pt;">[</span><span style="font-family: "arial" , sans-serif; font-size: 11pt;"><o:p></o:p></span></div>
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<b><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;">A)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; font-weight: normal; line-height: normal;"> </span></span></b><b><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;">FORMA ORDINARIA:<o:p></o:p></span></b></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">f(x) = a(x – h)<sup>2</sup> + k , coordenadas del vértice: V(h;k)<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Si a > 0, la parábola se abre hacia arriba.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Si a < 0, la parábola se abre hacia abajo.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , sans-serif; font-size: 11pt;">Ejemplo:<o:p></o:p></span></b></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Bosqueja su gráfica, calcula dominio y rango de f (x) = (x – 3)<sup>2</sup> + 1<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Coordenadas del vértice: V(3;1)<o:p></o:p></span></div>
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<a href="https://2.bp.blogspot.com/-6Ww01_W8FvM/XDUCqcL3aEI/AAAAAAAAEFQ/-HYVSNA-6nIkNpHB8ZwI-NtkrXvUBPmGwCLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="434" data-original-width="642" height="270" src="https://2.bp.blogspot.com/-6Ww01_W8FvM/XDUCqcL3aEI/AAAAAAAAEFQ/-HYVSNA-6nIkNpHB8ZwI-NtkrXvUBPmGwCLcBGAs/s400/A.png" width="400" /></a></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Dom(f) = R Rang(f) = </span><span style="font-family: "symbol"; font-size: 11pt;">[</span><span style="font-family: "arial" , sans-serif; font-size: 11pt;">1; +</span><span style="font-family: "symbol"; font-size: 11pt;">¥</span><span style="font-family: "symbol"; font-size: 11pt;">[</span><span style="font-family: "arial" , sans-serif; font-size: 11pt;"><o:p></o:p></span></div>
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<b><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;">A)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; font-weight: normal; line-height: normal;"> </span></span></b><b><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;">FORMA GENERAL:<o:p></o:p></span></b></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">F(x) = ax<sup>2</sup> + bx + c , para calcular las coordenadas del vértice, se lleva a la forma ordinaria, completando cuadrados.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , sans-serif; font-size: 11pt;">Ejemplo:<o:p></o:p></span></b></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Bosqueja su gráfica, calcula dominio y rango de f (x) = x <sup>2</sup> + 4x + 2<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Completando cuadrados se tiene f (x) = (x + 2)<sup>2</sup> – 2<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Coordenadas del vértice: V(-2;-2)<o:p></o:p></span></div>
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<a href="https://2.bp.blogspot.com/-Y_62anEf7lo/XDUEsrGvmoI/AAAAAAAAEFc/d6obFVg_bRU0ISMDnJJsDy6hYliYl0lcACLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="526" data-original-width="564" height="298" src="https://2.bp.blogspot.com/-Y_62anEf7lo/XDUEsrGvmoI/AAAAAAAAEFc/d6obFVg_bRU0ISMDnJJsDy6hYliYl0lcACLcBGAs/s320/A.png" width="320" /></a></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Dom(f) = R Rang(f) = </span><span style="font-family: "symbol"; font-size: 11pt;">[</span><span style="font-family: "arial" , sans-serif; font-size: 11pt;">-2; +</span><span style="font-family: "symbol"; font-size: 11pt;">¥</span><span style="font-family: "symbol"; font-size: 11pt;">[</span><span style="font-family: "arial" , sans-serif; font-size: 11pt;"><o:p></o:p></span></div>
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Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com2tag:blogger.com,1999:blog-1041179653668872995.post-73139843017078489812019-01-08T11:23:00.000-08:002019-01-08T11:29:30.198-08:00FUNCION CONSTANTE, FUNCION IDENTIDAD Y FUNCION LINEAL, GRAFICA, DOMINIO Y RANGO<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Blog de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/ox9aJg"><span style="color: blue;">https://goo.gl/ox9aJg</span></a></b></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Vídeos de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/PtnBkt" style="background-color: white;"><span style="color: blue;">https://goo.gl/PtnBkt</span></a></b></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Blog de matemática QuidiMat, teoría, ejemplos y problemas: </span><a href="https://goo.gl/wkR8Rc"><span style="color: blue;">https://goo.gl/wkR8Rc</span></a></b></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Tabulación de funciones con calculadora: </span><span style="color: blue;"><a href="https://youtu.be/Wv7uZf68Jlw">https://youtu.be/Wv7uZf68Jlw</a></span></b></span><br />
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/R38_FohTJPc/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/R38_FohTJPc?feature=player_embedded" width="320"></iframe></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><br /></b></span><span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b>Vídeo de funciones lineales, gráfica, dominio y rango: <a href="https://youtu.be/R38_FohTJPc">https://youtu.be/R38_FohTJPc</a></b></span><br />
<b style="color: red; font-family: arial, helvetica, sans-serif; text-indent: -22.65pt;"><span style="font-family: "arial" , sans-serif; font-size: 11pt;">a)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; font-weight: normal; line-height: normal;"> </span></span></b><b style="color: red; font-family: arial, helvetica, sans-serif; text-indent: -22.65pt;"><span style="font-family: "arial" , sans-serif; font-size: 11pt;">FUNCIÓN CONSTANTE</span></b><br />
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , sans-serif; font-size: 11pt;">A la función <b>f</b> le llamamos función constante, si su regla de correspondencia es:</span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , sans-serif; font-size: 11pt;">f(x) = c , donde c es una constante. Su dominio es Dom(f) = R, su rango es Rang(f) = </span><span style="font-family: "symbol"; font-size: 11pt;">{</span><span style="font-family: "arial" , sans-serif; font-size: 11pt;">c</span><span style="font-family: "symbol"; font-size: 11pt;">}</span><span style="font-family: "arial" , sans-serif; font-size: 11pt;">.<span style="color: red;"><o:p></o:p></span></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><a href="https://3.bp.blogspot.com/-zVFj0gLlEUA/XDT0ph4y8zI/AAAAAAAAEEI/4SYnSdkNSQMCJ_lrS8JzAY8H4csIrFoDwCLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="202" data-original-width="362" height="178" src="https://3.bp.blogspot.com/-zVFj0gLlEUA/XDT0ph4y8zI/AAAAAAAAEEI/4SYnSdkNSQMCJ_lrS8JzAY8H4csIrFoDwCLcBGAs/s320/A.png" width="320" /></a></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;"> <span style="color: red;">Ejemplo:</span></span></b></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , sans-serif; font-size: 11pt;"><b> </b>Construye su gráfica, calcula dominio y rango de f(x) = 3</span></span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;">Dom(f) = R, su rango es Rang(f) = </span><span lang="ES" style="font-family: "symbol"; font-size: 11pt;">{</span><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;">3</span><span lang="ES" style="font-family: "symbol"; font-size: 11pt;">}</span><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;">.</span></div>
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<b style="text-indent: -22.65pt;"><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;">b)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; font-weight: normal; line-height: normal;"> </span></span></b><b style="text-indent: -22.65pt;"><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;">FUNCIÓN IDENTIDAD</span></b></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">A la función f le llamaremos función identidad, si su regla de correspondencia es: f(x)=x o y=x. Su dominio es Dom(f) =R y su rango es Rang(f) = R.<o:p></o:p></span></div>
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<a href="https://4.bp.blogspot.com/-xWw19swZuYU/XDT1yPNH8nI/AAAAAAAAEEc/2W5wqvpf_7cJEHgLuRGtQ437W9GqoGwvgCLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="200" data-original-width="384" height="166" src="https://4.bp.blogspot.com/-xWw19swZuYU/XDT1yPNH8nI/AAAAAAAAEEc/2W5wqvpf_7cJEHgLuRGtQ437W9GqoGwvgCLcBGAs/s320/A.png" width="320" /></a></div>
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<b><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;"><span style="color: red;">Ejemplo:</span></span></b></div>
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;">Construye su gráfica, calcula dominio y rango de f(x) = x</span></span><br />
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 0cm;">Dom(f) = R </span><span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 0cm;">y su rango es Rang(f) = R.</span></div>
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<span style="text-indent: -22.65pt;"><span style="font-family: "arial" , sans-serif; font-size: 11pt;"><b><span style="color: red;">c</span></b></span><span style="font-family: "arial" , sans-serif; font-size: 11pt;"><b><span style="color: red;">)</span></b><span style="color: red; font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span></span><b style="text-indent: -22.65pt;"><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;">FUNCIÓN LINEAL</span></b></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 0cm;">A la función </span><b style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;">f</b><span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 0cm;"> le llamaremos función lineal, si su regla de correspondencia es:</span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 0cm;">f(x) = ax + b. Donde a y b son constantes y a </span><span style="font-family: "symbol"; font-size: 11pt; text-indent: 0cm;">¹</span><span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 0cm;"> 0.</span></div>
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<span style="height: 174px; margin-left: 181px; margin-top: 32px; position: absolute; width: 261px; z-index: 251653632;"></span><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;">También se expresa de la forma y = ax + b, donde su dominio es Dom(f) = R y </span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;">su rango es Rang(f) = R.</span></div>
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<b><span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;"><span style="color: red;">Ejemplo:</span></span></b></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt;">Construye su gráfica, calcula dominio y rango de f(x) = 2x -1</span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 0cm;">Dom(f) = R </span><span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 0cm;">y su rango es Rang(f) = R.</span></div>
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><br /></span>Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com0tag:blogger.com,1999:blog-1041179653668872995.post-19314734349894510602019-01-07T18:48:00.001-08:002019-01-07T18:48:15.240-08:00FUNCIONES: GRÁFICA, DOMINIO Y RANGO<div class="MsoNormal" style="text-align: center;">
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">Blog de funciones, gráfica, dominio y rango: <span style="line-height: 17.12px;"><a href="https://goo.gl/ox9aJg">https://goo.gl/ox9aJg</a><o:p></o:p></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">Vídeos de funciones, gráfica, dominio y rango: <span style="line-height: 17.12px;"><a href="https://goo.gl/PtnBkt">https://goo.gl/PtnBkt</a><o:p></o:p></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">Blog de matemática y funciones: <span style="line-height: 17.12px;"><a href="https://goo.gl/wkR8Rc">https://goo.gl/wkR8Rc</a><o:p></o:p></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">1) Función lineal, gráfica, dominio y rango: <a href="https://youtu.be/R38_FohTJPc">https://youtu.be/R38_FohTJPc</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">2) Función cuadrática, gráfica dominio y rango: <a href="https://youtu.be/bqM5L_Wxa4o">https://youtu.be/bqM5L_Wxa4o</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">3) Dominio de una función polinómica: <a href="https://youtu.be/3FKivJU6bbg">https://youtu.be/3FKivJU6bbg</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">4) Dominio de una función raíz de índice par: <a href="https://youtu.be/zsbc9JqRPto">https://youtu.be/zsbc9JqRPto</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">5) Dominio de una función raíz de índice impar: <a href="https://youtu.be/vbEnhXisC-g">https://youtu.be/vbEnhXisC-g</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">6) Dominio de una función racional: <a href="https://youtu.be/KX98MXnV96w">https://youtu.be/KX98MXnV96w</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">7) Rango de funciones, lineales, cuadráticas y racionales: <a href="https://youtu.be/-qdr2ncDSUY">https://youtu.be/-qdr2ncDSUY</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">8) Rango de una función racional: <a href="https://youtu.be/0lMR7K5GOlo">https://youtu.be/0lMR7K5GOlo</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">9) Rango de funciones con restricción: <a href="https://youtu.be/ROTDxVoEUmU">https://youtu.be/ROTDxVoEUmU</a></span><o:p></o:p><br />
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<b><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES DE R EN R - GRÁFICA, DOMINIO Y RANGO</span></u></b></div>
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<b style="text-indent: 0cm;"><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b style="text-indent: 0cm;"><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES</span></u></b></h4>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Uno de los más importantes conceptos de la matemática se refiere a un tipo especial de relaciones entre los elementos de dos conjuntos A y B, llamadas funciones de A en B.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Una <b>función </b>expresa la idea de una cantidad que depende de otra. Por ejemplo el área de un círculo depende de la medida de su radio, si se conoce la medida de la longitud del radio, su área está completamente determinada. Luego decimos que el área de un círculo es una función de la <b><i>longitud de su radio.</i></b><i><o:p></o:p></i></span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;">DEFINICIÓN DE FUNCIÓN<o:p></o:p></span></h5>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Una función es una correspondencia entre dos conjuntos, un conjunto de partida y un conjunto de llegada, tal que a cada valor del dominio le corresponde exactamente un valor del rango.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Una función de A en B es una relación f </span><span style="font-family: "symbol"; font-size: 11pt;"> </span><span style="font-family: "symbol"; font-size: 11pt;">Ì</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> A x B que hace corresponder a cada elemento del conjunto A a lo más un elemento y del conjunto B, se denota por y = f(x) </span><span style="font-family: "symbol"; font-size: 11pt;">Î</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> B. Al conjunto A se le llama conjunto de partida y al conjunto B conjunto de llegada.<o:p></o:p></span></div>
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<b><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">DOMINIO Y RANGO DE UNA FUNCIÓN<o:p></o:p></span></u></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Se llama dominio de una función f al conjunto de todos sus antecedentes (primeras componentes), y se denota por:<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Se llama rango o recorrido de la función f al conjunto de las imágenes de todos los elementos de A, vía f; y se denota Ran(f) ó R<sub>f</sub>.<o:p></o:p></span></div>
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<b><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">CÁLCULO DEL DOMINIO DE FUNCIONES USUALES<o:p></o:p></span></u></b></div>
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<b><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">a)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><span style="color: red;">FUNCIONES POLINÓMICAS -</span> <a href="https://youtu.be/3FKivJU6bbg" target="_blank">( Vídeo en YouTube )</a><o:p></o:p></span></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> </span><span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 19.85pt;">Es un polinomio de grado “n” con coeficientes racionales.</span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Dom </span><span style="font-family: "symbol"; font-size: 11pt;">[</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> P (x)</span><span style="font-family: "symbol"; font-size: 11pt;">]</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> = R, es decir, el dominio de cualquier función polinómica es el conjunto de números reales<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Ejemplo:</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Calcula el dominio de f(x) = x<sup>2</sup> – x +2<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Solución</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<span lang="ES" style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Dom f(x) = R</span></div>
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<span style="text-indent: -21.25pt;"><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;"><b>b)</b><span style="font-family: "times new roman"; font-size: 7pt;"> </span></span></span><b style="text-indent: -21.25pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><span style="color: red;">FUNCIONES CON RAÍCES DE ÍNDICE PAR -</span> <a href="https://youtu.be/zsbc9JqRPto" target="_blank">( Vídeo en YouTube)</a></span></b></div>
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<b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><br /></span></b><b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> <span style="color: red;">c)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></span></b><b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><span style="color: red;">FUNCIONES CON RAÍCES DE ÍNDICE IMPAR -</span> <a href="https://youtu.be/vbEnhXisC-g" target="_blank">( Vídeo en YouTube )</a></span></b></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Solución</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> En el numerador, “x” puede tomar cualquier valor real, pero, en el denominador “x” no puede tomar los valores de - 3 y 3.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Los diversos casos mencionados no siempre se presentan independientemente, es posible que en algunos ejercicios se presenten combinaciones de estos.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Ejemplo:</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Determina el dominio de </span></div>
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<b><u><span style="color: red;">ACTIVIDAD DE APRENDIZAJE Nro. 01<o:p></o:p></span></u></b></div>
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<span lang="ES" style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Determina el dominio de:<o:p></o:p></span></div>
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<span style="color: red;"><b>4</b></span><b style="text-indent: 0cm;"><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">.<span style="font-family: "times new roman"; font-size: 7pt;"> <span style="font-weight: normal;"> </span></span></span></b><b style="text-indent: 0cm;"><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">CÁLCULO DEL RANGO DE UNA FUNCIÓN</span></u></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Al determinar el rango de una función se presentan dos casos:<o:p></o:p></span></div>
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<b><u><span style="color: red;">CASO 1: (SIN RESTRICCIÓN)</span><o:p></o:p></u></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Cuando la función está definida en todo su dominio, entonces es suficiente despejar la variable “x” en términos de “y”= f(x), para luego analizar ¿Qué valores reales toma “y” de tal manera que “x” también sea real?. Como si se tratará del cálculo del dominio, pero esta vez, se trabaja con “y”.<o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><u>Vídeos de rango de funciones en YouTube</u></b></span></div>
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<span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><a href="https://youtu.be/-qdr2ncDSUY" target="_blank">Rango de funciones lineales, cuadráticas y racionales</a></b></span></div>
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<span style="color: blue; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><a href="https://youtu.be/ROTDxVoEUmU" target="_blank">Rango de funciones con restricción</a></b></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><u><span style="color: red;">CASO 2: (CON RESTRICCIÓN) -</span> <a href="https://youtu.be/ROTDxVoEUmU" target="_blank"><span style="color: red;">Vídeo</span></a><o:p></o:p></u></b></span></div>
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<u><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b>ACTIVIDAD DE APRENDIZAJE Nro. 2</b></span></u></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 35.45pt;">Calcula el rango de cada una de las funciones:</span></div>
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<b style="font-size: 11pt; text-indent: 0cm;"><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">5.<span style="font-family: "times new roman"; font-size: 7pt;"> </span></span><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">GRÁFICA DE UNA FUNCIÓN</span></b></div>
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<span lang="ES" style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Para graficar una función se realizan los siguientes pasos:<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">a)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Cálculo del dominio y rango de la función<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">b)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Determinación de las asíntotas, verticales y horizontales.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">c)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Tabulación, asignar valores a “x” para obtener otros valores en “y”.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">d)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Mapeo, es ubicar los puntos en el plano cartesiano.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">e)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Trazo de la curva, unir los puntos mediante curvas.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 35.45pt;"><b><u>ACTIVIDAD DE APRENDIZAJE Nro. 3<o:p></o:p></u></b></span></div>
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<b><a href="https://www.youtube.com/watch?v=R38_FohTJPc&t=4s&index=4&list=PLaV_odesVl_o96XjoIFGhdesMJ2BSlwv-" target="_blank"><span style="color: red;">Vídeo de gráfica, dominio y rango de una función lineal</span></a></b></div>
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Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com0tag:blogger.com,1999:blog-1041179653668872995.post-59754679953718487482018-06-29T11:09:00.000-07:002018-06-29T11:09:14.852-07:00CONCAVIDAD Y PUNTOS DE INFLEXION DE FUNCIONES GRAFICA Y ANALITICA – SEGUNDA DERIVADA<b style="font-family: arial, helvetica, sans-serif;"><span style="color: red;">Blog de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/ox9aJg"><span style="color: blue;">https://goo.gl/ox9aJg</span></a></b><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Vídeos de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/PtnBkt"><span style="color: blue;">https://goo.gl/PtnBkt</span></a><span style="color: red;"><o:p></o:p></span></b></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Blog de matemática QuidiMat, teoría, ejemplos y problemas: </span><span style="color: blue;"><span style="color: blue;"><a href="https://goo.gl/wkR8Rc">https://goo.gl/wkR8Rc</a></span><span style="color: red;"><o:p></o:p></span></span></b></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b>Concavidad y puntos de inflexión aplicando segunda derivada: <a href="https://youtu.be/CNXjDWfR_os">https://youtu.be/CNXjDWfR_os</a></b></span></div>
Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com0tag:blogger.com,1999:blog-1041179653668872995.post-12891942078497776502018-06-21T21:12:00.002-07:002018-06-24T10:24:31.203-07:00FUNCIONES CRECIENTES DECRECIENTES CON PRIMERA DERIVADA Y GRÁFICAMENTE<b style="font-family: arial, helvetica, sans-serif;"><span style="color: red;">Blog de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/ox9aJg"><span style="color: blue;">https://goo.gl/ox9aJg</span></a></b><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Vídeos de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/PtnBkt"><span style="color: blue;">https://goo.gl/PtnBkt</span></a><span style="color: red;"><o:p></o:p></span></b></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Blog de matemática QuidiMat, teoría, ejemplos y problemas: </span><span style="color: blue;"><span style="color: blue;"><a href="https://goo.gl/wkR8Rc">https://goo.gl/wkR8Rc</a></span><span style="color: red;"><o:p></o:p></span></span></b></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b>Funciones crecientes decrecientes derivando y gráfica: <a href="https://youtu.be/XVpxU_Z-j6U">https://youtu.be/kEf8O9Te49</a></b></span><br />
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b>Funciones crecientes y decrecientes (2): <a href="https://youtu.be/4-THSczFD8w">https://youtu.be/4-THSczFD8w</a></b></span><br />
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Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com0tag:blogger.com,1999:blog-1041179653668872995.post-80096823986068386532018-06-19T18:54:00.000-07:002018-06-19T18:54:14.214-07:00FUNCIONES POLINOMICAS GRAFICA DOMINIO RANGO APLICANDO DERIVADAS<b style="font-family: arial, helvetica, sans-serif;"><span style="color: red;">Blog de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/ox9aJg"><span style="color: blue;">https://goo.gl/ox9aJg</span></a></b><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Vídeos de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/PtnBkt"><span style="color: blue;">https://goo.gl/PtnBkt</span></a><span style="color: red;"><o:p></o:p></span></b></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Blog de matemática QuidiMat, teoría, ejemplos y problemas: </span><span style="color: blue;"><span style="color: blue;"><a href="https://goo.gl/wkR8Rc">https://goo.gl/wkR8Rc</a></span><span style="color: red;"><o:p></o:p></span></span></b></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b>Funciones polinómicas, gráfica, dominio y rango: <a href="https://youtu.be/kEf8O9Te490">https://youtu.be/kEf8O9Te49</a></b></span></div>
Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com0tag:blogger.com,1999:blog-1041179653668872995.post-73464351504097491422018-06-17T18:21:00.002-07:002019-01-08T12:17:18.607-08:00FUNCIONES CUADRATICAS, GRÁFICA, DOMINIO Y RANGO<b style="font-family: arial, helvetica, sans-serif;"><span style="color: red;">Blog de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/ox9aJg"><span style="color: blue;">https://goo.gl/ox9aJg</span></a></b><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Vídeos de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/PtnBkt"><span style="color: blue;">https://goo.gl/PtnBkt</span></a><span style="color: red;"><o:p></o:p></span></b></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Blog de matemática QuidiMat, teoría, ejemplos y problemas: </span><span style="color: blue;"><a href="https://goo.gl/wkR8Rc">https://goo.gl/wkR8Rc</a></span><span style="color: red;"><o:p></o:p></span></b></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b>Funciones cuadráticas, gráfica, dominio y rango: <a href="https://youtu.be/Mt2_eebcRWY">https://youtu.be/Mt2_eebcRWY</a></b></span><br />
<b style="text-indent: -22.65pt;"><span style="color: red; font-family: "Arial",sans-serif; font-size: 11.0pt; mso-fareast-font-family: Arial;">a)<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-weight: normal; line-height: normal;"> </span></span></b><b style="text-indent: -22.65pt;"><span style="color: red; font-family: "Arial",sans-serif; font-size: 11.0pt;">FUNCIÓN CUADRÁTICA</span></b><br />
<span style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 21.25pt;"> A la función f, le
llamaremos función cuadrática, si su regla de correspondencia es:</span><br />
<div class="MsoBodyTextFirstIndent2" style="margin-left: 42.1pt; text-indent: 21.25pt;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">f(x)= ax<sup>2</sup>+bx+c
donde x </span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">Î</span><span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"> R.</span></div>
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<b><span style="color: red; font-family: Arial, sans-serif; font-size: 11pt; text-indent: -18pt;">A)<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;">
</span></span><span style="color: red; font-family: Arial, sans-serif; font-size: 11pt; text-indent: -18pt;">FORMA CANÓNICA:</span></b></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">f(x) = x<sup>2</sup>
coordenadas del vértice: V(0;0)<o:p></o:p></span></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><b>Ejemplo:<o:p></o:p></b></span></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Bosqueja su gráfica,
calcula dominio y rango de f(x) = x<sup>2</sup><o:p></o:p></span></div>
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<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><sup><br /></sup></span></div>
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<a href="https://2.bp.blogspot.com/-uTeKO-3ORPs/XDUBhhr0iCI/AAAAAAAAEFE/flFZZX5HzRYo3BUxwAg1U4je-pVwT5wggCLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="486" data-original-width="630" height="246" src="https://2.bp.blogspot.com/-uTeKO-3ORPs/XDUBhhr0iCI/AAAAAAAAEFE/flFZZX5HzRYo3BUxwAg1U4je-pVwT5wggCLcBGAs/s320/A.png" width="320" /></a></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Dom(f) = R Rang(f) = </span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">[</span><span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">0;
+</span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">¥</span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">[</span><span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><o:p></o:p></span></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><b><span style="color: red; font-family: "Arial",sans-serif; font-size: 11.0pt; mso-fareast-font-family: Arial;">A)<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-weight: normal; line-height: normal;"> </span></span></b><!--[endif]--><b><span style="color: red; font-family: "Arial",sans-serif; font-size: 11.0pt;">FORMA ORDINARIA:<o:p></o:p></span></b></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">f(x) = a(x – h)<sup>2</sup>
+ k , coordenadas del vértice: V(h;k)<o:p></o:p></span></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Si a > 0, la parábola
se abre hacia arriba.<o:p></o:p></span></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Si a < 0, la parábola
se abre hacia abajo.<o:p></o:p></span></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<b><span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Ejemplo:<o:p></o:p></span></b></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Bosqueja su gráfica,
calcula dominio y rango de f (x) = (x – 3)<sup>2</sup> + 1<o:p></o:p></span></div>
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</div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Coordenadas del vértice:
V(3;1)<o:p></o:p></span></div>
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<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-6Ww01_W8FvM/XDUCqcL3aEI/AAAAAAAAEFQ/-HYVSNA-6nIkNpHB8ZwI-NtkrXvUBPmGwCLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="434" data-original-width="642" height="270" src="https://2.bp.blogspot.com/-6Ww01_W8FvM/XDUCqcL3aEI/AAAAAAAAEFQ/-HYVSNA-6nIkNpHB8ZwI-NtkrXvUBPmGwCLcBGAs/s400/A.png" width="400" /></a></div>
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<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Dom(f) = R Rang(f) = </span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">[</span><span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">1;
+</span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">¥</span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">[</span><span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><o:p></o:p></span></div>
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<br /></div>
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<!--[if !supportLists]--><b><span style="color: red; font-family: "Arial",sans-serif; font-size: 11.0pt; mso-fareast-font-family: Arial;">A)<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-weight: normal; line-height: normal;"> </span></span></b><!--[endif]--><b><span style="color: red; font-family: "Arial",sans-serif; font-size: 11.0pt;">FORMA GENERAL:<o:p></o:p></span></b></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">F(x) = ax<sup>2</sup> +
bx + c , para calcular las coordenadas del vértice, se lleva a la forma
ordinaria, completando cuadrados.<o:p></o:p></span></div>
<div class="MsoBodyTextFirstIndent2">
<br /></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<b><span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Ejemplo:<o:p></o:p></span></b></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Bosqueja su gráfica,
calcula dominio y rango de f (x) = x <sup>2</sup> + 4x + 2<o:p></o:p></span></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Completando cuadrados
se tiene f (x) = (x + 2)<sup>2</sup> – 2 <o:p></o:p></span></div>
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</div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Coordenadas del vértice:
V(-2;-2)<o:p></o:p></span></div>
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<a href="https://2.bp.blogspot.com/-Y_62anEf7lo/XDUEsrGvmoI/AAAAAAAAEFc/d6obFVg_bRU0ISMDnJJsDy6hYliYl0lcACLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="526" data-original-width="564" height="298" src="https://2.bp.blogspot.com/-Y_62anEf7lo/XDUEsrGvmoI/AAAAAAAAEFc/d6obFVg_bRU0ISMDnJJsDy6hYliYl0lcACLcBGAs/s320/A.png" width="320" /></a></div>
<div class="MsoBodyTextFirstIndent2" style="margin-left: 81.35pt; text-indent: 0cm;">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Dom(f) = R Rang(f) = </span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">[</span><span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">-2;
+</span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">¥</span><span style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">[</span><span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><o:p></o:p></span></div>
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<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><sup><br /></sup></span></div>
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<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><sup><br /></sup></span></div>
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<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><sup><br /></sup></span></div>
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<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><sup><br /></sup></span></div>
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<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><sup><br /></sup></span></div>
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Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com0tag:blogger.com,1999:blog-1041179653668872995.post-80283809821157649162018-06-15T21:52:00.002-07:002019-01-08T11:21:38.489-08:00FUNCION CONSTANTE, FUNCION IDENTIDAD Y FUNCION LINEAL, GRAFICA, DOMINIO Y RANGO<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Blog de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/ox9aJg"><span style="color: blue;">https://goo.gl/ox9aJg</span></a></b></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Vídeos de funciones, gráfica, dominio y rango: </span><a href="https://goo.gl/PtnBkt" style="background-color: white;"><span style="color: blue;">https://goo.gl/PtnBkt</span></a></b></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Blog de matemática QuidiMat, teoría, ejemplos y problemas: </span><a href="https://goo.gl/wkR8Rc"><span style="color: blue;">https://goo.gl/wkR8Rc</span></a></b></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><span style="color: red;">Tabulación de funciones con calculadora: </span><span style="color: blue;"><a href="https://youtu.be/Wv7uZf68Jlw">https://youtu.be/Wv7uZf68Jlw</a></span></b></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><br /></b></span>
<br />
<div class="separator" style="clear: both; text-align: left;">
</div>
<div style="text-align: center;">
<iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/R38_FohTJPc/0.jpg" src="https://www.youtube.com/embed/R38_FohTJPc?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
<br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><b><br /></b></span>
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b>Vídeo de funciones lineales, gráfica, dominio y rango: <a href="https://youtu.be/R38_FohTJPc">https://youtu.be/R38_FohTJPc</a></b></span><br />
<b style="color: red; font-family: arial, helvetica, sans-serif; text-indent: -22.65pt;"><span style="font-family: Arial, sans-serif; font-size: 11pt;">a)<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-weight: normal; line-height: normal;"> </span></span></b><b style="color: red; font-family: arial, helvetica, sans-serif; text-indent: -22.65pt;"><span style="font-family: Arial, sans-serif; font-size: 11pt;">FUNCIÓN CONSTANTE</span></b><br />
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">
</span><br />
<div class="MsoListContinue2">
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: Arial, sans-serif; font-size: 11pt;">A
la función <b>f</b> le llamamos función
constante, si su regla de correspondencia es:</span></span></div>
<div class="MsoBodyTextFirstIndent2">
<!--[endif]--><span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: Arial, sans-serif; font-size: 11pt;">f(x) = c , donde c es
una constante. Su dominio es <span style="mso-spacerun: yes;"> </span>Dom(f) = R,
su rango es Rang(f) = </span><span style="font-family: Symbol; font-size: 11pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">{</span></span><span style="font-family: Arial, sans-serif; font-size: 11pt;">c</span><span style="font-family: Symbol; font-size: 11pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">}</span></span><span style="font-family: Arial, sans-serif; font-size: 11pt;">.<span style="color: red;"><o:p></o:p></span></span></span></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><a href="https://3.bp.blogspot.com/-zVFj0gLlEUA/XDT0ph4y8zI/AAAAAAAAEEI/4SYnSdkNSQMCJ_lrS8JzAY8H4csIrFoDwCLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="202" data-original-width="362" height="178" src="https://3.bp.blogspot.com/-zVFj0gLlEUA/XDT0ph4y8zI/AAAAAAAAEEI/4SYnSdkNSQMCJ_lrS8JzAY8H4csIrFoDwCLcBGAs/s320/A.png" width="320" /></a></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><b><span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt;"> <span style="color: red;">Ejemplo:</span></span></b></span></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: Arial, sans-serif; font-size: 11pt;"><span style="color: red;"><b> </b></span>Construye su gráfica, calcula dominio y
rango de f(x) = 3</span></span></div>
<br />
<div class="MsoNormal" style="margin-left: 35.45pt;">
<span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-6e2JH_Eqbdc/XDT056HDzEI/AAAAAAAAEEQ/21Vov2FAjCcNCy_TgaUXBTf0uJmgihIUACLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="601" data-original-width="1357" height="176" src="https://4.bp.blogspot.com/-6e2JH_Eqbdc/XDT056HDzEI/AAAAAAAAEEQ/21Vov2FAjCcNCy_TgaUXBTf0uJmgihIUACLcBGAs/s400/A.png" width="400" /></a></div>
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><br /></span>
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"></span><br />
<div class="MsoNormal" style="margin-left: 35.45pt;">
<span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt;">Dom(f) = R, su rango es Rang(f) = </span><span lang="ES" style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">{</span><span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt;">3</span><span lang="ES" style="font-family: Symbol; font-size: 11.0pt; mso-ascii-font-family: Arial; mso-bidi-font-family: Arial; mso-char-type: symbol; mso-hansi-font-family: Arial; mso-symbol-font-family: Symbol;">}</span><span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt;">.</span></div>
<div class="MsoNormal" style="margin-left: 35.45pt;">
<br /></div>
<div class="MsoNormal" style="margin-left: 35.45pt;">
<span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt;">
</span></div>
<div class="MsoBodyTextFirstIndent2">
<b style="text-indent: -22.65pt;"><span style="color: red; font-family: Arial, sans-serif; font-size: 11pt;">b)<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-weight: normal; line-height: normal;"> </span></span></b><b style="text-indent: -22.65pt;"><span style="color: red; font-family: Arial, sans-serif; font-size: 11pt;">FUNCIÓN IDENTIDAD</span></b></div>
<div class="MsoBodyTextFirstIndent2">
<span style="font-family: "Arial",sans-serif; font-size: 11.0pt;">A la función f le llamaremos función identidad, si su regla
de correspondencia es: f(x)=x o y=x. Su dominio es Dom(f) =R y su rango es Rang(f)
= R.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 35.45pt;">
<span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-xWw19swZuYU/XDT1yPNH8nI/AAAAAAAAEEc/2W5wqvpf_7cJEHgLuRGtQ437W9GqoGwvgCLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="200" data-original-width="384" height="166" src="https://4.bp.blogspot.com/-xWw19swZuYU/XDT1yPNH8nI/AAAAAAAAEEc/2W5wqvpf_7cJEHgLuRGtQ437W9GqoGwvgCLcBGAs/s320/A.png" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
<b><span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><span style="color: red;">Ejemplo:</span></span></b></div>
<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">
<span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">Construye su gráfica, calcula dominio y rango de
f(x) = x</span></span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-tjuNsz2F654/XDT2Hm1zvDI/AAAAAAAAEEk/O9HEpX7BS2MkFJgpYGJwfOXbYxMLXU_gQCLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="601" data-original-width="1357" height="176" src="https://4.bp.blogspot.com/-tjuNsz2F654/XDT2Hm1zvDI/AAAAAAAAEEk/O9HEpX7BS2MkFJgpYGJwfOXbYxMLXU_gQCLcBGAs/s400/A.png" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;">Dom(f) = R </span><span style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;">y su rango es Rang(f) = R.</span></div>
<div class="separator" style="clear: both; text-align: left; text-indent: 0px;">
<span style="text-indent: -22.65pt;"><span style="font-family: Arial, sans-serif; font-size: 11pt;"><b><span style="color: red;">c</span></b></span><span style="font-family: Arial, sans-serif; font-size: 11pt;"><b><span style="color: red;">)</span></b><span style="color: red; font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; font-weight: normal; line-height: normal;"> </span></span></span><b style="text-indent: -22.65pt;"><span style="color: red; font-family: "Arial",sans-serif; font-size: 11.0pt;">FUNCIÓN LINEAL</span></b></div>
<div class="separator" style="clear: both; text-align: left; text-indent: 0px;">
<span style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;">A la función </span><b style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;">f</b><span style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;"> le llamaremos función lineal, si su
regla de correspondencia es:</span></div>
<div class="separator" style="clear: both; text-align: left; text-indent: 0px;">
<span style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;">f(x) = ax + b. Donde a
y b son constantes y a </span><span style="font-family: Symbol; font-size: 11pt; text-indent: 0cm;">¹</span><span style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;"> 0.</span></div>
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= ax + b, donde su dominio es Dom(f) = R y </span></div>
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<span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt; mso-ansi-language: ES; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES;">su rango es Rang(f) = R.</span></div>
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<a href="https://1.bp.blogspot.com/-rwO4aXT73AQ/XDT2-L0lsoI/AAAAAAAAEEw/XezS2C54BjIpac9X3X73ONdm69dh8gT6QCLcBGAs/s1600/A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="177" data-original-width="334" height="169" src="https://1.bp.blogspot.com/-rwO4aXT73AQ/XDT2-L0lsoI/AAAAAAAAEEw/XezS2C54BjIpac9X3X73ONdm69dh8gT6QCLcBGAs/s320/A.png" width="320" /></a></div>
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<b><span lang="ES" style="font-family: "Arial",sans-serif; font-size: 11.0pt;"><span style="color: red;">Ejemplo:</span></span></b></div>
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<span style="font-family: Arial, sans-serif; font-size: 11pt;">Construye su gráfica, calcula dominio y
rango de f(x) = 2x -1</span></div>
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<span style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;">Dom(f) = R </span><span style="font-family: Arial, sans-serif; font-size: 11pt; text-indent: 0cm;">y su rango es Rang(f) = R.</span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><br /></span>Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com1tag:blogger.com,1999:blog-1041179653668872995.post-81615395506679314802017-10-03T05:47:00.001-07:002017-10-03T05:47:58.201-07:00MATEMATICA FUNCIONES: GRÁFICA, DOMINIO, RANGO, FUNCION POLINOMICA, RAIZ INDICE PAR IMPAR RACIONAL<div class="MsoNormal" style="text-align: center;">
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span style="color: red;">Blog de funciones, gráfica, dominio y rango: </span><span style="line-height: 17.12px;"><a href="https://goo.gl/ox9aJg"><span style="color: blue;">https://goo.gl/ox9aJg</span></a><span style="color: red;"><o:p></o:p></span></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span style="color: red;">Vídeos de funciones, gráfica, dominio y rango: </span><span style="line-height: 17.12px;"><a href="https://goo.gl/PtnBkt"><span style="color: blue;">https://goo.gl/PtnBkt</span></a><span style="color: red;"><o:p></o:p></span></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;"><span style="color: red;">Blog de matemática y funciones: </span><span style="line-height: 17.12px;"><a href="https://goo.gl/wkR8Rc"><span style="color: blue;">https://goo.gl/wkR8Rc</span></a><span style="color: red;"><o:p></o:p></span></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">1) Función lineal, gráfica, dominio y rango: <a href="https://youtu.be/R38_FohTJPc">https://youtu.be/R38_FohTJPc</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">2) Función cuadrática, gráfica dominio y rango: <a href="https://youtu.be/bqM5L_Wxa4o">https://youtu.be/bqM5L_Wxa4o</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">3) Dominio de una función polinómica: <a href="https://youtu.be/3FKivJU6bbg">https://youtu.be/3FKivJU6bbg</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">4) Dominio de una función raíz de índice par: <a href="https://youtu.be/zsbc9JqRPto">https://youtu.be/zsbc9JqRPto</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">5) Dominio de una función raíz de índice impar: <a href="https://youtu.be/vbEnhXisC-g">https://youtu.be/vbEnhXisC-g</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">6) Dominio de una función racional: <a href="https://youtu.be/KX98MXnV96w">https://youtu.be/KX98MXnV96w</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">7) Rango de funciones, lineales, cuadráticas y racionales: <a href="https://youtu.be/-qdr2ncDSUY">https://youtu.be/-qdr2ncDSUY</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">8) Rango de una función racional: <a href="https://youtu.be/0lMR7K5GOlo">https://youtu.be/0lMR7K5GOlo</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">9) Rango de funciones con restricción: <a href="https://youtu.be/ROTDxVoEUmU">https://youtu.be/ROTDxVoEUmU</a></span><o:p></o:p></div>
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<b><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES DE R EN R - GRÁFICA, DOMINIO Y RANGO</span></u></b></div>
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<b style="text-indent: 0cm;"><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b style="text-indent: 0cm;"><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES</span></u></b></h4>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Uno de los más importantes conceptos de la matemática se refiere a un tipo especial de relaciones entre los elementos de dos conjuntos A y B, llamadas funciones de A en B.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Una <b>función </b>expresa la idea de una cantidad que depende de otra. Por ejemplo el área de un círculo depende de la medida de su radio, si se conoce la medida de la longitud del radio, su área está completamente determinada. Luego decimos que el área de un círculo es una función de la <b><i>longitud de su radio.</i></b><i><o:p></o:p></i></span></div>
<h5 style="text-indent: 35.45pt;">
<span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;">DEFINICIÓN DE FUNCIÓN<o:p></o:p></span></h5>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Una función es una correspondencia entre dos conjuntos, un conjunto de partida y un conjunto de llegada, tal que a cada valor del dominio le corresponde exactamente un valor del rango.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Una función de A en B es una relación f </span><span style="font-family: "symbol"; font-size: 11pt;"> </span><span style="font-family: "symbol"; font-size: 11pt;">Ì</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> A x B que hace corresponder a cada elemento del conjunto A a lo más un elemento y del conjunto B, se denota por y = f(x) </span><span style="font-family: "symbol"; font-size: 11pt;">Î</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> B. Al conjunto A se le llama conjunto de partida y al conjunto B conjunto de llegada.<o:p></o:p></span></div>
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<a href="https://1.bp.blogspot.com/-m1wZCsHk00c/VtdjkwGXM3I/AAAAAAAABfE/BneQb1WWXxA/s1600/Sin%2Bt%25C3%25ADtulo.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-m1wZCsHk00c/VtdjkwGXM3I/AAAAAAAABfE/BneQb1WWXxA/s1600/Sin%2Bt%25C3%25ADtulo.png" /></a></div>
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<b><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">DOMINIO Y RANGO DE UNA FUNCIÓN<o:p></o:p></span></u></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Se llama dominio de una función f al conjunto de todos sus antecedentes (primeras componentes), y se denota por:<o:p></o:p></span></div>
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<a href="https://3.bp.blogspot.com/-N8Q2JxgIymw/VtdkNcCRKnI/AAAAAAAABfM/HehzdGdlii0/s1600/Sin%2Bt%25C3%25ADtulo.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-N8Q2JxgIymw/VtdkNcCRKnI/AAAAAAAABfM/HehzdGdlii0/s1600/Sin%2Bt%25C3%25ADtulo.png" /></a></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Se llama rango o recorrido de la función f al conjunto de las imágenes de todos los elementos de A, vía f; y se denota Ran(f) ó R<sub>f</sub>.<o:p></o:p></span></div>
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<b><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">CÁLCULO DEL DOMINIO DE FUNCIONES USUALES<o:p></o:p></span></u></b></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">a)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES POLINÓMICAS - <a href="https://www.youtube.com/watch?v=3FKivJU6bbg&index=7&list=PLaV_odesVl_o96XjoIFGhdesMJ2BSlwv-" target="_blank">( Vídeo en YouTube )</a><o:p></o:p></span></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> </span><span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 19.85pt;">Es un polinomio de grado “n” con coeficientes racionales.</span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Dom </span><span style="font-family: "symbol"; font-size: 11pt;">[</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> P (x)</span><span style="font-family: "symbol"; font-size: 11pt;">]</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> = R, es decir, el dominio de cualquier función polinómica es el conjunto de números reales<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Ejemplo:</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Calcula el dominio de f(x) = x<sup>2</sup> – x +2<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Solución</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<span lang="ES" style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Dom f(x) = R</span></div>
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<span style="text-indent: -21.25pt;"><span style="font-family: "arial" , sans-serif; font-size: 11pt;"><b>b)</b><span style="font-family: "times new roman"; font-size: 7pt;"> </span></span></span><b style="text-indent: -21.25pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES CON RAÍCES DE ÍNDICE PAR - <a href="https://www.youtube.com/watch?v=zsbc9JqRPto&t=5s&index=2&list=PLaV_odesVl_o96XjoIFGhdesMJ2BSlwv-" target="_blank">( Vídeo en YouTube)</a></span></b></div>
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<b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> </span></b><br />
<b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><br /></span></b><b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> c)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES CON RAÍCES DE ÍNDICE IMPAR - <a href="https://www.youtube.com/watch?v=vbEnhXisC-g&index=6&list=PLaV_odesVl_o96XjoIFGhdesMJ2BSlwv-" target="_blank">( Vídeo en YouTube )</a></span></b><br />
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<b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> d</span></b><b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b style="text-indent: -8.55pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES RACIONALES - <a href="https://www.youtube.com/watch?v=KX98MXnV96w&index=1&list=PLaV_odesVl_o96XjoIFGhdesMJ2BSlwv-" target="_blank">( Vídeo en YouTUbe )</a></span></b><br />
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Solución</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> En el numerador, “x” puede tomar cualquier valor real, pero, en el denominador “x” no puede tomar los valores de - 3 y 3.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Los diversos casos mencionados no siempre se presentan independientemente, es posible que en algunos ejercicios se presenten combinaciones de estos.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Ejemplo:</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Determina el dominio de </span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Solución</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<b><u><span style="color: red;">ACTIVIDAD DE APRENDIZAJE Nro. 01<o:p></o:p></span></u></b></div>
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<span lang="ES" style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Determina el dominio de:<o:p></o:p></span></div>
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4<b style="text-indent: 0cm;"><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b style="text-indent: 0cm;"><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">CÁLCULO DEL RANGO DE UNA FUNCIÓN</span></u></b><br />
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Al determinar el rango de una función se presentan dos casos:<o:p></o:p></span></div>
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<b><u>CASO 1: (SIN RESTRICCIÓN)<o:p></o:p></u></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Cuando la función está definida en todo su dominio, entonces es suficiente despejar la variable “x” en términos de “y”= f(x), para luego analizar ¿Qué valores reales toma “y” de tal manera que “x” también sea real?. Como si se tratará del cálculo del dominio, pero esta vez, se trabaja con “y”.<o:p></o:p></span><br />
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<span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><u>Vídeos de rango de funciones en YouTube</u></b></span></div>
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<span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><a href="https://www.youtube.com/watch?v=-qdr2ncDSUY&index=8&list=PLaV_odesVl_o96XjoIFGhdesMJ2BSlwv-" target="_blank">Rango de funciones lineales, cuadráticas y racionales</a></b></span><br />
<span style="color: blue; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><a href="https://www.youtube.com/watch?v=ROTDxVoEUmU&index=3&list=PLaV_odesVl_o96XjoIFGhdesMJ2BSlwv-" target="_blank">Rango de funciones con restricción</a></b></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><u>CASO 2: (CON RESTRICCIÓN) - <a href="https://youtu.be/ROTDxVoEUmU" target="_blank"><span style="color: red;">Vídeo</span></a><o:p></o:p></u></b></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Cuando el dominio de la función se encuentra restringido. En este caso se pueden utilizar diversos métodos de resolución.</span><a href="https://1.bp.blogspot.com/-QrttELem2vI/VtdsZ1-rBfI/AAAAAAAABg8/h4m70w88rlA/s1600/Sin%2Bt%25C3%25ADtulo.png" imageanchor="1" style="clear: left; display: inline !important; font-size: 11pt; margin-bottom: 1em; margin-right: 1em; text-align: center; text-indent: 0cm;"><img border="0" src="https://1.bp.blogspot.com/-QrttELem2vI/VtdsZ1-rBfI/AAAAAAAABg8/h4m70w88rlA/s1600/Sin%2Bt%25C3%25ADtulo.png" /></a></span></div>
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<u><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><b>ACTIVIDAD DE APRENDIZAJE Nro. 2</b></span></u></div>
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<u><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><br /></b></span></u></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 35.45pt;">Calcula el rango de cada una de las funciones:</span></div>
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<b style="font-size: 11pt; text-indent: 0cm;"><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">5.<span style="font-family: "times new roman"; font-size: 7pt;"> </span></span><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">GRÁFICA DE UNA FUNCIÓN</span></b></div>
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<span lang="ES" style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Para graficar una función se realizan los siguientes pasos:<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">a)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Cálculo del dominio y rango de la función<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">b)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Determinación de las asíntotas, verticales y horizontales.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">c)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Tabulación, asignar valores a “x” para obtener otros valores en “y”.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">d)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Mapeo, es ubicar los puntos en el plano cartesiano.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">e)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Trazo de la curva, unir los puntos mediante curvas.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 35.45pt;"><b><u>ACTIVIDAD DE APRENDIZAJE Nro. 3<o:p></o:p></u></b></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 35.45pt;"><span lang="ES" style="font-family: "arial" , "sans-serif"; mso-bidi-font-size: 11.0pt;">Construye la gráfica de las siguientes funciones:<o:p></o:p></span></span></div>
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<b><a href="https://www.youtube.com/watch?v=R38_FohTJPc&t=4s&index=4&list=PLaV_odesVl_o96XjoIFGhdesMJ2BSlwv-" target="_blank"><span style="color: red;">Vídeo de gráfica, dominio y rango de una función lineal</span></a></b></div>
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Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com0tag:blogger.com,1999:blog-1041179653668872995.post-25511281127024509162017-06-30T21:24:00.002-07:002017-09-30T16:27:44.471-07:00FUNCIONES: DOMINIO, RANGO Y GRAFICA<div class="MsoNormal" style="text-align: center;">
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">Blog de funciones, gráfica, dominio y rango: <span style="line-height: 17.12px;"><a href="https://goo.gl/ox9aJg">https://goo.gl/ox9aJg</a><o:p></o:p></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">Vídeos de funciones, gráfica, dominio y rango: <span style="line-height: 17.12px;"><a href="https://goo.gl/PtnBkt">https://goo.gl/PtnBkt</a><o:p></o:p></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">Blog de matemática y funciones: <span style="line-height: 17.12px;"><a href="https://goo.gl/wkR8Rc">https://goo.gl/wkR8Rc</a><o:p></o:p></span></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">1) Función lineal, gráfica, dominio y rango: <a href="https://youtu.be/R38_FohTJPc">https://youtu.be/R38_FohTJPc</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">2) Función cuadrática, gráfica dominio y rango: <a href="https://youtu.be/bqM5L_Wxa4o">https://youtu.be/bqM5L_Wxa4o</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">3) Dominio de una función polinómica: <a href="https://youtu.be/3FKivJU6bbg">https://youtu.be/3FKivJU6bbg</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">4) Dominio de una función raíz de índice par: <a href="https://youtu.be/zsbc9JqRPto">https://youtu.be/zsbc9JqRPto</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">5) Dominio de una función raíz de índice impar: <a href="https://youtu.be/vbEnhXisC-g">https://youtu.be/vbEnhXisC-g</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">6) Dominio de una función racional: <a href="https://youtu.be/KX98MXnV96w">https://youtu.be/KX98MXnV96w</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">7) Rango de funciones, lineales, cuadráticas y racionales: <a href="https://youtu.be/-qdr2ncDSUY">https://youtu.be/-qdr2ncDSUY</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">8) Rango de una función racional: <a href="https://youtu.be/0lMR7K5GOlo">https://youtu.be/0lMR7K5GOlo</a><o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "helvetica" , sans-serif;">9) Rango de funciones con restricción: <a href="https://youtu.be/ROTDxVoEUmU">https://youtu.be/ROTDxVoEUmU</a></span><o:p></o:p></div>
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<b><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES DE R EN R - GRÁFICA, DOMINIO Y RANGO</span></u></b></div>
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<b style="text-indent: 0cm;"><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b style="text-indent: 0cm;"><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">FUNCIONES</span></u></b></h4>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Uno de los más importantes conceptos de la matemática se refiere a un tipo especial de relaciones entre los elementos de dos conjuntos A y B, llamadas funciones de A en B.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Una <b>función </b>expresa la idea de una cantidad que depende de otra. Por ejemplo el área de un círculo depende de la medida de su radio, si se conoce la medida de la longitud del radio, su área está completamente determinada. Luego decimos que el área de un círculo es una función de la <b><i>longitud de su radio.</i></b><i><o:p></o:p></i></span></div>
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<span lang="ES" style="font-family: "arial" , sans-serif; font-size: 11pt;">DEFINICIÓN DE FUNCIÓN<o:p></o:p></span></h5>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Una función es una correspondencia entre dos conjuntos, un conjunto de partida y un conjunto de llegada, tal que a cada valor del dominio le corresponde exactamente un valor del rango.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Una función de A en B es una relación f </span><span style="font-family: "symbol"; font-size: 11pt;"> </span><span style="font-family: "symbol"; font-size: 11pt;">Ì</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> A x B que hace corresponder a cada elemento del conjunto A a lo más un elemento y del conjunto B, se denota por y = f(x) </span><span style="font-family: "symbol"; font-size: 11pt;">Î</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> B. Al conjunto A se le llama conjunto de partida y al conjunto B conjunto de llegada.<o:p></o:p></span></div>
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<b><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">DOMINIO Y RANGO DE UNA FUNCIÓN<o:p></o:p></span></u></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Se llama dominio de una función f al conjunto de todos sus antecedentes (primeras componentes), y se denota por:<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Se llama rango o recorrido de la función f al conjunto de las imágenes de todos los elementos de A, vía f; y se denota Ran(f) ó R<sub>f</sub>.<o:p></o:p></span></div>
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<b><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">CÁLCULO DEL DOMINIO DE FUNCIONES USUALES<o:p></o:p></span></u></b></div>
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<b><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">a)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><span style="color: red;">FUNCIONES POLINÓMICAS -</span> <a href="https://youtu.be/3FKivJU6bbg" target="_blank">( Vídeo en YouTube )</a><o:p></o:p></span></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> </span><span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 19.85pt;">Es un polinomio de grado “n” con coeficientes racionales.</span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Dom </span><span style="font-family: "symbol"; font-size: 11pt;">[</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> P (x)</span><span style="font-family: "symbol"; font-size: 11pt;">]</span><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> = R, es decir, el dominio de cualquier función polinómica es el conjunto de números reales<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Ejemplo:</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Calcula el dominio de f(x) = x<sup>2</sup> – x +2<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Solución</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<span lang="ES" style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Dom f(x) = R</span></div>
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<span style="text-indent: -21.25pt;"><span style="color: red; font-family: "arial" , sans-serif; font-size: 11pt;"><b>b)</b><span style="font-family: "times new roman"; font-size: 7pt;"> </span></span></span><b style="text-indent: -21.25pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><span style="color: red;">FUNCIONES CON RAÍCES DE ÍNDICE PAR -</span> <a href="https://youtu.be/zsbc9JqRPto" target="_blank">( Vídeo en YouTube)</a></span></b></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Solución</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> En el numerador, “x” puede tomar cualquier valor real, pero, en el denominador “x” no puede tomar los valores de - 3 y 3.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Los diversos casos mencionados no siempre se presentan independientemente, es posible que en algunos ejercicios se presenten combinaciones de estos.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Ejemplo:</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Determina el dominio de </span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Solución</span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><o:p></o:p></span></div>
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<b><u><span style="color: red;">ACTIVIDAD DE APRENDIZAJE Nro. 01<o:p></o:p></span></u></b></div>
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<span lang="ES" style="font-family: "arial" , "sans-serif"; font-size: 11pt;"> Determina el dominio de:<o:p></o:p></span></div>
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<span style="color: red;"><b>4</b></span><b style="text-indent: 0cm;"><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">.<span style="font-family: "times new roman"; font-size: 7pt;"> <span style="font-weight: normal;"> </span></span></span></b><b style="text-indent: 0cm;"><u><span lang="ES" style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">CÁLCULO DEL RANGO DE UNA FUNCIÓN</span></u></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Al determinar el rango de una función se presentan dos casos:<o:p></o:p></span></div>
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<b><u><span style="color: red;">CASO 1: (SIN RESTRICCIÓN)</span><o:p></o:p></u></b></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Cuando la función está definida en todo su dominio, entonces es suficiente despejar la variable “x” en términos de “y”= f(x), para luego analizar ¿Qué valores reales toma “y” de tal manera que “x” también sea real?. Como si se tratará del cálculo del dominio, pero esta vez, se trabaja con “y”.<o:p></o:p></span></div>
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<span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><u>Vídeos de rango de funciones en YouTube</u></b></span></div>
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<span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><a href="https://youtu.be/-qdr2ncDSUY" target="_blank">Rango de funciones lineales, cuadráticas y racionales</a></b></span></div>
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<span style="color: blue; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><a href="https://youtu.be/ROTDxVoEUmU" target="_blank">Rango de funciones con restricción</a></b></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><b><u><span style="color: red;">CASO 2: (CON RESTRICCIÓN) -</span> <a href="https://youtu.be/ROTDxVoEUmU" target="_blank"><span style="color: red;">Vídeo</span></a><o:p></o:p></u></b></span></div>
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<span style="font-family: "arial" , "sans-serif"; font-size: 11pt;"><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Cuando el dominio de la función se encuentra restringido. En este caso se pueden utilizar diversos métodos de resolución.</span><a href="https://1.bp.blogspot.com/-QrttELem2vI/VtdsZ1-rBfI/AAAAAAAABg8/h4m70w88rlA/s1600/Sin%2Bt%25C3%25ADtulo.png" imageanchor="1" style="clear: left; display: inline !important; font-size: 11pt; margin-bottom: 1em; margin-right: 1em; text-align: center; text-indent: 0cm;"><img border="0" src="https://1.bp.blogspot.com/-QrttELem2vI/VtdsZ1-rBfI/AAAAAAAABg8/h4m70w88rlA/s1600/Sin%2Bt%25C3%25ADtulo.png" style="cursor: move;" /></a></span></div>
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<u><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;"><b>ACTIVIDAD DE APRENDIZAJE Nro. 2</b></span></u></div>
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<b style="font-size: 11pt; text-indent: 0cm;"><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">5.<span style="font-family: "times new roman"; font-size: 7pt;"> </span></span><span style="color: red; font-family: "arial" , "sans-serif"; font-size: 11pt;">GRÁFICA DE UNA FUNCIÓN</span></b></div>
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<span lang="ES" style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Para graficar una función se realizan los siguientes pasos:<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">a)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Cálculo del dominio y rango de la función<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">b)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Determinación de las asíntotas, verticales y horizontales.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">c)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Tabulación, asignar valores a “x” para obtener otros valores en “y”.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">d)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Mapeo, es ubicar los puntos en el plano cartesiano.<o:p></o:p></span></div>
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<b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">e)<span style="font-family: "times new roman"; font-size: 7pt; font-weight: normal;"> </span></span></b><span style="font-family: "arial" , "sans-serif"; font-size: 11pt;">Trazo de la curva, unir los puntos mediante curvas.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 35.45pt;"><b><u>ACTIVIDAD DE APRENDIZAJE Nro. 3<o:p></o:p></u></b></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 11pt; text-indent: 35.45pt;"><span lang="ES" style="font-family: "arial" , "sans-serif"; mso-bidi-font-size: 11.0pt;">Construye la gráfica de las siguientes funciones:<o:p></o:p></span></span></div>
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<b><a href="https://www.youtube.com/watch?v=R38_FohTJPc&t=4s&index=4&list=PLaV_odesVl_o96XjoIFGhdesMJ2BSlwv-" target="_blank"><span style="color: red;">Vídeo de gráfica, dominio y rango de una función lineal</span></a></b></div>
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Vídeos de Matemática – QuidiMat - Guillermohttp://www.blogger.com/profile/10932452086049554061noreply@blogger.com0