Chapter EP, Problem 7.1.12EP

To determine

To calculate: The value of $\left[g\circ h\right]\left(x\right)\text{ and }\left[h\circ g\right]\left(x\right)$ where $g\left(x\right)=3x+2\text{ and }h\left(x\right)=5-3x$ .

Answer to Problem 7.1.12EP

The value of $\left[g\circ h\right]\left(x\right)=17-9x\text{ and }\left[h\circ g\right]\left(x\right)=-1-9x$ .

Explanation of Solution

Given information:

The functions $g\left(x\right)=3x+2\text{ and }h\left(x\right)=5-3x$ .

Formula used:

The composition of two functions f and gis $\left[f\circ g\right]\left(x\right)=f\left(g\left(x\right)\right)$ .

Calculation:

Consider the functions $g\left(x\right)=3x+2\text{ and }h\left(x\right)=5-3x$ .

Recall that the composition of two functions f and gis $\left[f\circ g\right]\left(x\right)=f\left(g\left(x\right)\right)$ .

Therefore, $\left[g\circ h\right]\left(x\right)\text{ and }\left[h\circ g\right]\left(x\right)$ are evaluated as,

  $\begin{array}{c}\left[g\circ h\right]\left(x\right)=g\left(h\left(x\right)\right)\\ =g\left(5-3x\right)\\ =3\left(5-3x\right)+2\\ =15-9x+2\end{array}$

Simplify it further as,

  $\begin{array}{c}\left[g\circ h\right]\left(x\right)=15-9x+2\\ =17-9x\end{array}$

And,

  $\begin{array}{c}\left[h\circ g\right]\left(x\right)=h\left(g\left(x\right)\right)\\ =h\left(3x+2\right)\\ =5-3\left(3x+2\right)\\ =5-9x-6\end{array}$

Simplify it further as,

  $\begin{array}{c}\left[h\circ g\right]\left(x\right)=5-9x-6\\ =-1-9x\end{array}$

Thus, the value of $\left[g\circ h\right]\left(x\right)=17-9x\text{ and }\left[h\circ g\right]\left(x\right)=-1-9x$ .