Question

20. Si los puntos (-4,2) y (3,-5) pertenecen a la gráfica de la función lineal f, entonces el criterio de la función inversa de f es:
a. f-1(x)=-x+2
b. f-1x=x+6
c. f-1(x)=-x-2
d. f-1(x)=x+2

Answer 1

Antes de determinar la inversa lo primero que realizaremos será hallar la función que pase por aquellos puntos, entonces

                                $\begin{array}{c}\boxed{\:\:\boldsymbol{\mathsf{\vphantom{\rule{1pt}{20pt}_{\rule{1pt}{17pt}}}y-y_1 = \left(\dfrac{y_2-y_1}{x_2-x_1}\right)(x - x_1)}}\:\:}\\\\\mathsf{Siendo\ (x_1,y_1)\ y \ (x_2,y_2)\ los\ puntos}\end{array}$

En el problema tenemos que:

               $\begin{array}{cccccccccc}\blue{\boldsymbol{\bigcirc \kern-8pt \triangleright}}\:\:\:\:\mathsf{\boldsymbol{\mathsf{A=}(}\:\overbrace{\boldsymbol{-4}}^{x_1}\:\boldsymbol{,}\:\underbrace{\boldsymbol{2}}_{y_1}\:\boldsymbol{)}}&&&&&&&&&\blue{\boldsymbol{\bigcirc \kern-8pt \triangleright}\:\:\:\:}\mathsf{\boldsymbol{\mathsf{B=}(}\:\overbrace{\boldsymbol{3}}^{x_2}\:\boldsymbol{,}\:\underbrace{\boldsymbol{-5}}_{y_2}\:\boldsymbol{)}}\end{array}$

Reemplazamos

                                      $\begin{array}{c}\mathsf{y-y_1 = \left(\dfrac{y_2-y_1}{x_2-x_1}\right)(x - x_1)}\\\\\mathsf{y-(2) = \left(\dfrac{-5-(2)}{3-(-4)}\right)\big(x - (-4)\big)}\\\\\mathsf{y - 2 = \left(\dfrac{-5 -2}{3+4}\right)\big(x + 4\big)}\\\\\mathsf{y-2 = \left(\dfrac{-7}{\not\!7}\right)\big(x + 4\big)}\\\\\mathsf{(y-2) = (-1)(x + 4)}\\\\\mathsf{y-2= -x - 4}\\\\\red{\boxed{\boxed{\boldsymbol{\mathsf{y = -x-2}}}}}\end{array}$

Despejaremos "x" para determinar su inversa

                                     $\begin{array}{c}\\\sf{y=-x - 2}\\\\\sf{x=-y - 2}\\\\\sf{Cambiamos\ x\ por \ f^{-1}(x)e\ y\ por\ x\\ \sf{y\ tendremos\ la\ inversa\ de\ la\ funci\acute{o}n}\\\\\boxed{\boxed{\boldsymbol{\red{\sf{f^{-1}(x) = -x-2}}}}}\end{array}$

Rpta. Alternativa c.

                                              $\boxed{\sf{{R}}\quad\raisebox{10pt}{ \sf{\red{O}} }\!\!\!\!\raisebox{-10pt}{ \sf{\red{O}} }\quad\raisebox{15pt}{ \sf{{G}} }\!\!\!\!\raisebox{-15pt}{ \sf{{G}} }\quad\raisebox{15pt}{ \sf{\red{H}} }\!\!\!\!\raisebox{-15pt}{ \sf{\red{H}} }\quad\raisebox{10pt}{ \sf{{E}} }\!\!\!\!\raisebox{-10pt}{ \sf{{E}} }\quad\sf{\red{R}}}\hspace{-64.5pt}\rule{10pt}{.2ex}\:\rule{3pt}{1ex}\rule{3pt}{1.5ex}\rule{3pt}{2ex}\rule{3pt}{1.5ex}\rule{3pt}{1ex}\:\rule{10pt}{.2ex}$