Chapter 6.1, Problem 31PPS

To determine

If $f\left(x\right)=x^2+6x-2\text{ and }g\left(x\right)=x-6$ then find the value of $\left[f\circ g\right]\left(x\right)\text{ and }\left[g\circ f\right]\left(x\right)$ if they exist.

Answer to Problem 31PPS

The value of $\left[f\circ g\right]\left(x\right)=x^2-6x-2\text{and }\left[g\circ f\right]\left(x\right)=x^2+6x-8$ .

Explanation of Solution

Given information: The given functions are,

  $\begin{array}{l}f\left(x\right)=x^2+6x-2\\ g\left(x\right)=x-6\end{array}$

Calculation: Evaluating $\left[f\circ g\right]\left(x\right)$ ,

  $\begin{array}{c}\left[f\circ g\right]\left(x\right)=f\left[g\left(x\right)\right]\\ =f\left(x-6\right)\\ ={\left(x-6\right)}^2+6\left(x-6\right)-2\\ =x^2-6x-2\end{array}$

Similarly, evaluating $\left[g\circ f\right]\left(x\right)$ ,

  $\begin{array}{c}\left[g\circ f\right]\left(x\right)=g\left[f\left(x\right)\right]\\ =g\left(x^2+6x-2\right)\\ =\left(x^2+6x-2\right)-6\\ =x^2+6x-8\end{array}$