Chapter 6.2, Problem 31PPS

To determine

Whether the pair of functions $f\left(x\right)=\frac{1}{2}x+5\text{ and }g\left(x\right)=2x-10$ are inverse functions or not.

Answer to Problem 31PPS

Yes, the given pair of functions $f\left(x\right)=\frac{1}{2}x+5\text{ and }g\left(x\right)=2x-10$ are inverse functions.

Explanation of Solution

Given information: The given pair of functions are,

  $\left(\begin{array}{l}f\left(x\right)=\frac{1}{2}x+5\\ g\left(x\right)=2x-10\end{array}\right)\mathrm{......}\left(1\right)$

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),

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left[g\left(x\right)\right]\\ =f\left(2x-10\right)\\ =\frac{1}{2}\left(2x-10\right)+5\\ =x\mathrm{....}\left(\text{which is a identity function}\right)\end{array}$

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

  $\begin{array}{c}\left(g\circ f\right)\left(x\right)=g\left[f\left(x\right)\right]\\ =g\left(\frac{1}{2}x+5\right)\\ =2\left(\frac{1}{2}x+5\right)-10\\ =x\mathrm{....}\left(\text{also a identity function}\right)\end{array}$

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 $f\left(x\right)=\frac{1}{2}x+5\text{ and }g\left(x\right)=2x-10$ are inverse functions.