Chapter 7.3, Problem 60STP

To determine

Whether the given pair of functions are inverse functions.

Answer to Problem 60STP

Yes

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=\frac{3x+2}{5}\\ g(x)=\frac{5x-2}{3}\end{array}$

Concept Used :

 A function g is the inverse of a function f if whenever

  $\begin{array}{l}y=f(x)\text{ then }x=g(y)\\ \text{That is ,}\\ \text{f and g are inverse functions if :}\\ \left[f\circ g\right](x)=x\\ and\\ \left[g\circ f\right](x)=x\end{array}$

Calculation:

  $\begin{array}{l}\text{f and g are inverse functions if and only if :}\\ \left[f\circ g\right](x)=x\\ and\\ \left[g\circ f\right](x)=x\\ \\ Given\\ f(x)=\frac{3x+2}{5}\\ g(x)=\frac{5x-2}{3}\\ \\ Check\\ \left[f\circ g\right](x)=f\left[g(x)\right]=f\left(\frac{5x-2}{3}\right)=\frac{3\cdot \left(\frac{5x-2}{3}\right)+2}{5}=x\\ \left[g\circ f\right](x)=g\left[f(x)\right]=g\left(\frac{3x+2}{5}\right)=\frac{5\cdot \left(\frac{3x+2}{5}\right)-2}{3}=x\\ \\ So,\\ f\&g\text{ are inverse functions}\text{.}\end{array}$