Chapter 6.4, Problem 10MCQ

To determine

Whether the pair of functions $f\left(x\right)=x^2-5\text{ and }g\left(x\right)=5+x^{-2}$ are inverse functions or not. Write yes or no.

Answer to Problem 10MCQ

No, the given pair of functions $f\left(x\right)=x^2-5\text{ and }g\left(x\right)=5+x^{-2}$ are not inverse functions.

Explanation of Solution

Given information: The given pair of functions are,

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

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 are 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),

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left[g\left(x\right)\right]\\ =f\left(5+x^{-2}\right)\\ ={\left(5+x^{-2}\right)}^2-5\\ =20+x^{-4}+10x^{-2}\mathrm{....}\left(\begin{array}{l}\text{ which isn't a }\\ \text{identity function}\end{array}\right)\end{array}$

Since $\left(f\circ g\right)\left(x\right)$ is not an identity function therefore the given pair of functions $f\left(x\right)=x^2-5\text{ and }g\left(x\right)=5+x^{-2}$ are not inverse functions.