Chapter 1.3, Problem 59E

To determine

To check: Whether $f\circ g$ is linear if f and g are linear; find the slope of the graph $f\circ g$ if it is linear.

Answer to Problem 59E

The composition function $f\circ g$ is a linear function of the slope $m_1{m}_2$.

Explanation of Solution

Given:

The linear functions are $f\left(x\right)=m_{1}x+b_1,g\left(x\right)=m_{2}x+b_2$.

Calculation:

The composite function $\left(f\circ g\right)\left(x\right)$ is defined as follows.

$\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$

Substitute $m_{2}x+b_2$ for x in $f\left(x\right)$ and find the composition function $\left(f\circ g\right)\left(x\right)$.

$\begin{array}{c}f\left(g\left(x\right)\right)=m_1\left(m_{2}x+b_2\right)+b_1\\ =m_1{m}_{2}x+m_1{b}_2+b_1\end{array}$

Observe that $\left(f\circ g\right)\left(x\right)$ in the form of the linear function $y=mx+c$ where m is slope and c is constant.

Thus, it can be concluded that the composite function $\left(f\circ g\right)\left(x\right)$ is a linear function with the slope $m_1{m}_2$.