Question

find $(f\circ g)\left(x\right)$and$(g\circ f)\left(x\right)$and graph each of these functions.

localid="1646342074607" role="math" $$\begin{gathered} f\left(x\right)=\tan x \\ g\left(x\right)=4x \end{gathered}$$

Short answer

If $f\left(x\right)=\tan x$and $g\left(x\right)=4x$then$(f\circ g)\left(x\right)=\tan \left(4x\right)$and its graph is

and $(g\circ f)\left(x\right)=4\tan \left(x\right)$and its graph is

Step-by-step solution

Step 1. Given data

The given functions are

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

Step 2. Composition of function

$(f\circ g)\left(x\right)$composition of function is

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

Consider different values of x and determine the coordinate of several points

Step 3. Graph of (f∘g)(x)

Plot the points of$(f\circ g)\left(x\right)=\tan \left(4x\right)$and connect them smoothly

Step 4. Composition of function

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

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

Consider different values of x and determine the coordinate of several points

Step 5. Graph of (g∘f)(x)

Plot the points of$(g\circ f)\left(x\right)=4\tan \left(x\right)$ and connect them smoothly