Chapter 4.7, Problem 35PPS

To determine

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

Answer to Problem 35PPS

The equation of inverse function is $f^{-1}(x)=\frac{1}{4}x+\frac{9}{2}$

Explanation of Solution

Given information:Slope of f(x) given 4. Also given, $f^{-1}(5)=2$ .

Calculation:

Slope of f(x) = 4.

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 $f^{-1}(x)$ = $\frac{1}{4}$ .

Given, $f^{-1}(5)=2$ .

That means the point $(2,5)$ lies on the function $f^{-1}(x)$ .

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 = $\frac{1}{4}$ , the point $(x_1,y_1)$ = $(2,5)$ .

  $\therefore$ The equation of the inverse function is

  $\begin{array}{l}y-y_1=m(x-x_1)\\ \Rightarrow y-5=\frac{1}{4}(x-2)\\ \Rightarrow y-5=\frac{1}{4}x-\frac{2}{4}\\ \Rightarrow y=\frac{1}{4}x-\frac{1}{2}+5\\ \Rightarrow y=\frac{1}{4}x-\frac{1}{2}+\frac{10}{2}\\ \Rightarrow y=\frac{1}{4}x+\frac{10-1}{2}\\ \Rightarrow y=\frac{1}{4}x+\frac{9}{2}\end{array}$

  $\therefore$ The inverse function is $f^{-1}(x)=\frac{1}{4}x+\frac{9}{2}$