Chapter 8.2, Problem 69S

To determine

To find: The expressions $\left[g\circ h\right]\left(x\right)$ , and $\left[h\circ g\right]\left(x\right)$ .

Answer to Problem 69S

The value of $\left[g\circ h\right]\left(x\right)=x^2-1$ , and $\left[h\circ g\right]\left(x\right)=x^2-6x+11$ .

Explanation of Solution

Given:

The functions are $h\left(x\right)=x^2+2$ , and $g\left(x\right)=x-3$ .

Calculation:

Find the composite function $\left[g\circ h\right]\left(x\right)$ as follows.

  $\begin{array}{c}\left[g\circ h\right]\left(x\right)=g\left(h\left(x\right)\right)\\ =g\left(x^2+2\right)\\ =x^2+2-3\\ =x^2-1\end{array}$

Find the composite function $\left[h\circ g\right]\left(x\right)$ as follows.

  $\begin{array}{c}\left[h\circ g\right]\left(x\right)=h\left(g\left(x\right)\right)\\ =h\left(x-3\right)\\ ={\left(x-3\right)}^2+2\\ =x^2-6x+9+2\end{array}$

Simplify further.

  $\left[h\circ g\right]\left(x\right)=x^2-6x+11$

Therefore, the value of $\left[g\circ h\right]\left(x\right)=x^2-1$ , and $\left[h\circ g\right]\left(x\right)=x^2-6x+11$ .