Chapter 3.4, Problem 10Q

Solution

Verify that f and g are inverse functions.

  $f^{-1}(x)=g(x)$ so f and g are inverse function.

Given:

  $f(x)=5x^3$

  $g(x)=\sqrt[3]{\frac{x}{5}}$

Concept Used:

• In order to find the inverse of a function $y=f\left(x\right)$, follow below steps:
• Step 1: interchange x and y in the given function
• Step 2: solve for y and that will give the required inverse function

Calculation:

  $f(x)=5x^3$

  $g(x)=\sqrt[3]{\frac{x}{5}}$

The inverse of $f(x)=5x^3$

  $\begin{array}{c}f(x)=5x^3\\ y=f(x)\\ y=5x^3\\ solveforx\\ y=5x^3\\ x^3=\frac{y}{5}\\ x=\sqrt[3]{\frac{y}{5}}\\ swapxwithy\\ y=\sqrt[3]{\frac{x}{5}}\\ f^{-1}(x)=\sqrt[3]{\frac{x}{5}}\end{array}$

So the inverse of $f(x)=5x^3$ is $f^{-1}(x)=\sqrt[3]{\frac{x}{5}}$

So $f^{-1}(x)=g(x)$ that means f and g are inverse function.