Q. 84 - Precalculus Enhanced with Graphing Utilities

Question

find $(f\circ g)\left(x\right)$and $(g\circ f)\left(x\right)$and graph each of these functions.
$$\begin{gathered} f\left(x\right)=-3x \\ g\left(x\right)=\sin x \end{gathered}$$

Step-by-Step Solution

Answer

If $f\left(x\right)=-3x and g\left(x\right)=\sin x$then $(g\circ f)\left(x\right)=-\sin \left(3x\right) and (f\circ g)\left(x\right)=-3 \sin \left(x\right)$and their graph is 

Step 1. Given data

The given functions are 

$$\begin{gathered} f\left(x\right)=-3x \\ g\left(x\right)=\sin x \end{gathered}$$

Step 2. compositions of functions

Composition $(f\circ g)\left(x\right)$is

$$\begin{gathered} (f\circ g)\left(x\right)=f\left(g\right(x\left)\right) \\ =f\left(\sin \right(x\left)\right) \\ =-3 \sin x \end{gathered}$$

Step 3. compositions of functions

Composition $(g\circ f)\left(x\right)$is

$$\begin{gathered} (g\circ f)\left(x\right)=g\left(f\right(x\left)\right) \\ =g(-3x) \\ =\sin (-3x) \\ =-\sin \left(3x\right) \end{gathered}$$

Step 4. Graph of the composition of functions

Plot the graph of $(f\circ g)\left(x\right)=-3 \sin \left(x\right)$and $(g\circ f)\left(x\right)=-\sin \left(3x\right)$