Q. 82 - 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.
$$\begin{gathered} f\left(x\right)=\cos x \\ g\left(x\right)=\frac{1}{2}x \end{gathered}$$

Step-by-Step Solution

Answer

If $f\left(x\right)=\cos x \& g\left(x\right)=\frac{1}{2}x$then$(g\circ f)\left(x\right)=\frac{1}{2}\cos \left(x\right) \& (f\circ g)\left(x\right)=\cos \left(\frac{1}{2}x\right)$ and their graph is 

Step 1. Given data

The given functions are 

$$\begin{gathered} f\left(x\right)=\cos x \\ g\left(x\right)=\frac{1}{2}x \end{gathered}$$

Step 2. compositions of functions

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

$$\begin{gathered} (f\circ g)\left(x\right)=f\left(g\right(x\left)\right) \\ =f\left(\frac{1}{2}x\right) \\ =\cos \left(\frac{1}{2}x\right) \end{gathered}$$

Step 3. compositions of functions

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

$$\begin{gathered} (g\circ f)\left(x\right)=g\left(f\right(x\left)\right) \\ =g\left(\cos \right(x\left)\right) \\ =\frac{1}{2}\cos x \end{gathered}$$

Step 4. Graph of the composition of functions

Plot the graph of$(f\circ g)\left(x\right)=\cos \left(\frac{1}{2}x\right)$ and$(g\circ f)\left(x\right)=\frac{1}{2}\cos x$