Chapter 6.2, Problem 35PPS

To determine

Whether the pair of functions $f\left(x\right)=x^2-9\text{ and }g\left(x\right)=x+3$ are inverse functions or not.

Answer to Problem 35PPS

No, the given pair of functions $f\left(x\right)=x^2-9\text{ and }g\left(x\right)=x+3$ aren’t inverse functions.

Explanation of Solution

Given information: The given pair of functions are,

  $\left(\begin{array}{l}f\left(x\right)=x^2-9\\ g\left(x\right)=x+3\end{array}\right)\mathrm{......}\left(1\right)$

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),

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left[g\left(x\right)\right]\\ =f\left(x+3\right)\\ ={\left(x+3\right)}^2-9\\ =x^2+6x\mathrm{....}\left(\text{which isn't a identity function}\right)\end{array}$

Since $\left(f\circ g\right)\left(x\right)$ isn’t an identity function therefore the given pair of functions $f\left(x\right)=x^2-9\text{ and }g\left(x\right)=x+3$ aren’t inverse functions.