Chapter 5.6, Problem 23E

To determine

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

Answer to Problem 23E

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

Explanation of Solution

Given information:

the function is $f\left(x\right)=4x^2,x\le 0$ .

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

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

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

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

$x$	$f\left(x\right)$
$-1$	$4$
$0$	$0$
$-2$	$16$

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: