Chapter 4.7, Problem 32PPS

To determine

The equation of the inverse function $f^{-1}(x)$ .

Answer to Problem 32PPS

The equation of inverse function is $f^{-1}(x)=\frac{1}{7}x-\frac{6}{7}$

Explanation of Solution

Given information:

Slope of f(x) given 7. Also given, graph of $f^{-1}(x)$ contains the point $(13,1)$ .

Calculation:

Slope of f(x) = 7.

We know that slope of a function and it’s inverse function are multiplicative inverses of each other. That means slope of a function and inverse function are reciprocal of each other.

  $\therefore$ Slope of inverse function $f^{-1}(x)$ = $\frac{1}{7}$ .

Given, $f^{-1}(x)$ contains the point $(13,1)$

We know that, the equation of a line in point-slope form is $y-y_1=m(x-x_1)$ , where m is the slope and the point lies on the line is $(x_1,y_1)$ .

Here we have the slope m = $\frac{1}{7}$ , the point $(x_1,y_1)$ = $(13,1)$ .

  $\therefore$ The equation of the inverse function is

  $\begin{array}{l}y-y_1=m(x-x_1)\\ \Rightarrow y-1=\frac{1}{7}(x-13)\\ \Rightarrow y-1=\frac{1}{7}x-\frac{13}{7}\\ \Rightarrow y=\frac{1}{7}x-\frac{13}{7}+1\\ \Rightarrow y=\frac{1}{7}x-\frac{13}{7}+\frac{7}{7}\\ \Rightarrow y=\frac{1}{7}x+\frac{7-13}{7}\\ \Rightarrow y=\frac{1}{7}x-\frac{6}{7}\end{array}$

  $\therefore$ The inverse function is $f^{-1}(x)=\frac{1}{7}x-\frac{6}{7}$