Chapter 1, Problem 19CT

To determine

To state : whether the function has an inverse function.

Answer to Problem 19CT

Yes, function has an inverse function and the inverse function is $g\left(x\right)={\left(\frac{8x}{3}\right)}^{\frac{2}{3}}$.

Explanation of Solution

Given information :

The function is $f\left(x\right)=\frac{3x\sqrt{x}}{8}$.

from the above function it can be observed that the function has an inverse function because for every value of $y$ there is an unique x -value.

For inverse function,

  $\begin{array}{l}y=\frac{3x\sqrt{x}}{8}\\ 8y=3x\sqrt{x}\text{ [multiply by 8]}\\ x^{\frac{3}{2}}=\frac{8y}{3}\text{ [divide by 3]}\\ x={\left(\frac{8y}{3}\right)}^{\frac{2}{3}}\text{ [root]}\end{array}$

The inverse function is $g\left(x\right)={\left(\frac{8x}{3}\right)}^{\frac{2}{3}}$.