Chapter 6.2, Problem 28PPS

To determine

Whether the pair of functions $f\left(x\right)=4x+6\text{ and }g\left(x\right)=\frac{x-6}{4}$ are inverse functions or not.

Answer to Problem 28PPS

Yes, the given pair of functions $f\left(x\right)=4x+6\text{ and }g\left(x\right)=\frac{x-6}{4}$ are inverse functions.

Explanation of Solution

Given information: The given pair of functions are,

  $\left(\begin{array}{l}f\left(x\right)=4x+6\\ g\left(x\right)=\frac{x-6}{4}\end{array}\right)\mathrm{......}\left(1\right)$

Concept used:

The two functions $f\left(x\right)\text{ and }g\left(x\right)$ are inverse to each other if and only if both of their compositions result identity function. Mathematically,

  $\left(f\circ g\right)x=\left(g\circ f\right)x=x$

Calculation: Evaluating $\left(f\circ g\right)\left(x\right)$ using (1),

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left[g\left(x\right)\right]\\ =f\left(\frac{x-6}{4}\right)\\ =4\left(\frac{x-6}{4}\right)+6\\ =x\mathrm{....}\left(\text{which is a identity function}\right)\end{array}$

Similarly, evaluating $\left(g\circ f\right)\left(x\right)$ using (1),

  $\begin{array}{c}\left(g\circ f\right)\left(x\right)=g\left[f\left(x\right)\right]\\ =g\left(4x+6\right)\\ =\frac{\left(4x+6\right)-6}{4}\\ =x\mathrm{....}\left(\text{also a identity function}\right)\end{array}$

Since $\left(f\circ g\right)\left(x\right)=\left(g\circ f\right)\left(x\right)$ and both are identity functions therefore the given pair of functions $f\left(x\right)=4x+6\text{ and }g\left(x\right)=\frac{x-6}{4}$ are inverse functions.