Chapter 5.6, Problem 19E

To determine

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

Answer to Problem 19E

The inverse function is $f^{-1}\left(x\right)=\frac{3}{2}x+2$ .

Explanation of Solution

Given information:

the function is $f\left(x\right)=\frac{2}{3}x-\frac{1}{3}$ .

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

  $\begin{array}{l}f\left(x\right)=\frac{2}{3}x-\frac{1}{3}\\ y=\frac{2}{3}x-\frac{1}{3}\text{ [substitute }f\left(x\right)=y]\\ \frac{2}{3}x=y+\frac{1}{3}\text{ [add }\frac{1}{3}\text{ both side]}\\ x=\frac{3}{2}y+2\text{ [multiply and simplify]}\\ f^{-1}\left(x\right)=\frac{3}{2}x+2\text{ [simplify]}\end{array}$

Thus, the inverse is $f^{-1}\left(x\right)=\frac{3}{2}x+2$ .

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

$x$	$f\left(x\right)$
$3$	$\frac{5}{3}$
$0$	$-\frac{1}{3}$
$-3$	$-\frac{7}{3}$

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: