Q. 83 - 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.
$$\begin{gathered} f\left(x\right)=-2x \\ g\left(x\right)=\cos x \end{gathered}$$

Step-by-Step Solution

Answer

If$f\left(x\right)=-2x \& g\left(x\right)=\cos x$ then $(g\circ f)\left(x\right)=\cos (-2x) and (f\circ g)\left(x\right)=-2 \cos x$and their graph is

Step 1. Given data

The given functions are 

$$\begin{gathered} f\left(x\right)=-2x \\ g\left(x\right)=\cos x \end{gathered}$$

Step 2. compositions of functions

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

$$\begin{gathered} (f\circ g)\left(x\right)=f\left(g\right(x\left)\right) \\ =f\left(\cos \right(x\left)\right) \\ =-2 \cos x \end{gathered}$$

Step 3. compositions of functions

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

$$\begin{gathered} (g\circ f)\left(x\right)=g\left(f\right(x\left)\right) \\ =g(-2x) \\ =\cos (-2x) \end{gathered}$$

Step 4. Graph of the composition of functions

Plot the graph of $(f\circ g)\left(x\right)=-2 \cos x$and$(g\circ f)\left(x\right)=\cos (-2x)$