Chapter 8.2, Problem 71S

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 71S

The value of $\left[g\circ h\right]\left(x\right)=-15x-5$ , and $\left[h\circ g\right]\left(x\right)=-15x+25$ .

Explanation of Solution

Given:

The functions are $h\left(x\right)=-5x$ , and $g\left(x\right)=3x-5$ .

Calculation:

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

  $\begin{array}{c}\left[g\circ h\right]\left(x\right)=g\left(h\left(x\right)\right)\\ =g\left(-5x\right)\\ =3\left(-5x\right)-5\\ =-15x-5\end{array}$

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

  $\begin{array}{c}\left[h\circ g\right]\left(x\right)=h\left(g\left(x\right)\right)\\ =h\left(3x-5\right)\\ =-5\left(3x-5\right)\\ =-15x+25\end{array}$

Therefore, the value of $\left[g\circ h\right]\left(x\right)=-15x-5$ , and $\left[h\circ g\right]\left(x\right)=-15x+25$ .