Question

The graphs of the functions

$\begin{array}{l}f\left(x\right)=m_{1}x+b_1\\ g\left(x\right)=m_{2}x+b_2\end{array}$

are lines with slopes $m_1\text{and}{m}_2$ respectively. Is the graph of$f\circ g$a line? If so, what is the slope?

Short answer

The graph of $f\circ g$ is a linewith slope $m_1{m}_2$.

Step-by-step solution

Step 1. Apply the concept of slope-intercept form.

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Step 2. Apply the concept of composite function.

The composition of two functions fand g is expressed as $f\circ g$. This is also called composite function. Mathematically it is expressed below.

width="127">f∘gx=fgx

Step 3. Find composition of functions.

Consider the provided functions

$\begin{array}{c}f\left(x\right)=m_{1}x+b_1\\ g\left(x\right)=m_{2}x+b_2\end{array}$

Evaluate the composition of the functions $f\circ g$.

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

Rewrite the obtained expression as $\left(f\circ g\right)\left(x\right)=mx+c$where $m=m_1{m}_2\text{and}c=\left(m_1{b}_2+b_1\right)$

Where m is the slope and c is the intercept of y-axis.

Since, the composition function $f\circ g$can be written in slope intercept form which represent an equation of line.

Therefore, graph of $f\circ g$is a line.

Thus, the graph of $f\circ g$is a line with slope $m_1{m}_2$.