Chapter 6.2, Problem 53HP

To determine

To Find:

An example of a function and its inverse.

Answer to Problem 53HP

  $\begin{array}{l}f(x)=x\\ f^{-1}(x)=x\end{array}$

Explanation of Solution

Calculation:

  $\begin{array}{l}Conside\text{ the function :}\\ f(x)=x\\ \text{We find its inverse:}\\ \text{Replace f(x) by y:}\\ y=x\\ \text{Replace x by y and y by x:}\\ x=y\\ \text{Solve for y:}\\ y=x\\ So,\\ f^{-1}(x)=x\\ Verify:\\ \left[f\circ f^{-1}\right]\left(x\right)=f\left[f^{-1}(x)\right]=f(x)=x\\ \left[f^{-1}\circ f\right]\left(x\right)=f^{-1}\left[f(x)\right]=f^{-1}(x)=x\\ Hence,\text{ the two functions are inverse}\text{.}\end{array}$