Chapter 5.6, Problem 17E

To determine

Tofind: the inverse of the function and graph the function and inverse function.

Answer to Problem 17E

The inverse function is $f^{-1}\left(x\right)=8-2x$ .

Explanation of Solution

Given information:

the function is $f\left(x\right)=-\frac{1}{2}x+4$ .

Calculation: to find the inverse function, first substitute $y=f\left(x\right)$ then simplify,

  $\begin{array}{l}f\left(x\right)=-\frac{1}{2}x+4\\ y=-\frac{1}{2}x+4\text{ [substitute }f\left(x\right)=y]\\ -\frac{1}{2}x=y-4\text{ [subtract 4 from both side]}\\ x=8-2y\text{ [multiply and simplify]}\\ f^{-1}\left(x\right)=8-2x\text{ [simplify]}\end{array}$

Thus, the inverse is $f^{-1}\left(x\right)=8-2x$ .

To draw the graph of the function and inverse function, first use the plotting point method,

$x$	$f\left(x\right)$
$2$	$3$
$0$	$4$
$-2$	$5$

Now by jointing these points, the graph of function can be obtained and the graph of the inverse function is the reflection in the line $y=x$ .

Thus, the graph of function and inverse function can be observed as: