Chapter SH, Problem 6.5EP

To determine

Check whether the given pair of functions are inverse of each other or not.

Answer to Problem 6.5EP

The given pair of functions are inverse of each other.

Explanation of Solution

Given:

The given functions are $f\left(x\right)=2x-8$ and $g\left(x\right)=\frac{1}{2}x+4$ .

Calculation:

For the inverse function $\left(f\circ g\right)\left(x\right)=\left(g\circ f\right)\left(x\right)=x$ .

Apply property.

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=f\left[g\left(x\right)\right]\\ f\left(x\right)=2x-8\end{array}$

Substitute the value of $x=\frac{1}{2}x+4$ .

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)\\ =f\left[\frac{1}{2}x+4\right]\\ =2\left(\frac{1}{2}x+4\right)-8\\ =x+8-8\\ =x\end{array}$

Apply property.

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)=g\left[f\left(x\right)\right]\\ g\left(x\right)=\frac{1}{2}x+4\end{array}$

Substitute the value of $x=2x-8$ .

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)\\ =g\left[2x-8\right]\\ =\frac{1}{2}\left(2x-8\right)+4\\ =x-4+4\\ =x\end{array}$

Hence the given pair of functions are inverse of each other.