Chapter 6.3, Problem 58SR

To determine

Whether the given pair of functions are inverse functions.

Answer to Problem 58SR

Yes

Explanation of Solution

Given:

  $\begin{array}{l}f(x)=2x\\ g(x)=\frac{1}{2}x\end{array}$

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:

  $\begin{array}{l}\text{f and g are inverse functions if and only if :}\\ \left[f\circ g\right](x)=x\\ and\\ \left[g\circ f\right](x)=x\\ Given\\ f(x)=2x\\ g(x)=\frac{1}{2}x\\ Check\\ \left[f\circ g\right](x)=f\left[g(x)\right]=f\left(\frac{1}{2}x\right)=2\cdot \left(\frac{1}{2}x\right)=x\\ \left[g\circ f\right](x)=g\left[f(x)\right]=g(2x)=\frac{1}{2}\cdot (2x)=x\\ So,\\ f\&g\text{ are inverse functions}\text{.}\end{array}$