Q. 81 - 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.

$$\begin{gathered} f\left(x\right)=\sin x \\ g\left(x\right)=4x \end{gathered}$$

Step-by-Step Solution

Answer

If $f\left(x\right)=\sin x \& g\left(x\right)=4x$then $(f\circ g)\left(x\right)=\sin \left(4x\right)$and $(g\circ f)\left(x\right)=4 \sin \left(x\right)$their graph is

Step 1. Given data

The given functions are

$$\begin{gathered} f\left(x\right)=\sin x \\ g\left(x\right)=4x \end{gathered}$$

Step 2. compositions of functions

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

$$\begin{gathered} (f\circ g)\left(x\right)=f\left(g\right(x\left)\right) \\ =f\left(4x\right) \\ =\sin \left(4x\right) \end{gathered}$$

Step 3. composition of functions

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

$$\begin{gathered} (g\circ f)\left(x\right)=g\left(f\right(x\left)\right) \\ =g\left(\sin \right(x\left)\right) \\ =4 \sin \left(x\right) \end{gathered}$$

Step 4. Graph of the composition of functions

Plot the graph of $(f\circ g)\left(x\right)=\sin \left(4x\right)$ and$(g\circ f)\left(x\right)=4 \sin \left(x\right)$