Chapter 1.2, Problem 3CFU

To determine

whether $\left[f\circ g\right]\left(x\right)$ is always equal to $\left[g\circ f\right]\left(x\right)$ , explain the answer with examples and counterexamples.

Answer to Problem 3CFU

No

Explanation of Solution

Given information: Two functions are as follows:

  $f\left(x\right)$

  $g\left(x\right)$

Calculation:

  $\left[f\circ g\right]\left(x\right)$ is $f\left(x\right)$ performed on $g\left(x\right)$ i.e., $\left[f\circ g\right]\left(x\right)=f\left(g\left(x\right)\right)$ .

  $\left[g\circ f\right]\left(x\right)$ is $g\left(x\right)$ performed on $g\left(x\right)$ i.e., $\left[g\circ f\right]\left(x\right)=g\left(f\left(x\right)\right)$ .

Therefore, they are not always equal.

  $\begin{array}{c}f\left(g\left(x\right)\right)\ne g\left(f\left(x\right)\right)\\ \left[f\circ g\right]\left(x\right)\ne \left[g\circ f\right]\left(x\right)\end{array}$

Counterexample: let’s find the $\left[f\circ g\right]\left(x\right)$ and $\left[g\circ f\right]\left(x\right)$ for $f\left(x\right)=x^2-1\text{ and }g\left(x\right)=1$ .

  $\begin{array}{c}\left[f\circ g\right]\left(x\right)=f\left(g\left(x\right)\right)\\ =0\end{array}$

Similarly,

  $\begin{array}{c}\left[g\circ f\right]\left(x\right)=g\left(f\left(x\right)\right)\\ =1\end{array}$

Therefore, $\left[f\circ g\right]\left(x\right)\ne \left[g\circ f\right]\left(x\right)$ .