Chapter 1.6, Problem 80E

To determine

To find : the inverse function.

Answer to Problem 80E

The inverse function is $f^{-1}\left(x\right)=\frac{36}{x^2}$.

Explanation of Solution

Given information :

The function $f\left(x\right)=\frac{6}{\sqrt{x}}$.

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

  $\begin{array}{l}f(x)=\frac{6}{\sqrt{x}}\\ y=\frac{6}{\sqrt{x}}\text{ [substitute }f\left(x\right)=y]\\ \sqrt{x}=\frac{6}{y}\text{ [cross multiplication]}\\ x=\frac{6}{y^2}\text{ [square ]}\\ f^{-1}\left(x\right)=\frac{36}{x^2}\text{ [simplify]}\end{array}$

Thus, the inverse function is $f^{-1}\left(x\right)=\frac{36}{x^2}$.

Now by using graphing utility the graph of function $f\text{ and }{f}^{-1}$ can be obtained as:

  

Form the above graph it can be observed that the graph of function $f\text{ and }g$ are the reflection of each other in the line $y=x$.