Question

find $(f\circ g)\left(x\right)$and$(g\circ f)\left(x\right)$ and graph each of these functions.
role="math" localid="1646344152688" $$\begin{gathered} f\left(x\right)=-2x \\ g\left(x\right)=cotx \end{gathered}$$

Short answer

If $f\left(x\right)=-2x$and $g\left(x\right)=cotx$then$(f\circ g)\left(x\right)=-2cot\left(x\right)$and its graph is

and $(g\circ f)\left(x\right)=cot(-2x)$and its graph is

Step-by-step solution

Step 1. Given data

The given functions are

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

Step 2. Composition of function 

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

role="math" localid="1646344262300" $$\begin{gathered} (f\circ g)\left(x\right)=f\left(g\right(x\left)\right) \\ (f\circ g)\left(x\right)=f(cot(x\left)\right) \\ (f\circ g)\left(x\right)=-2(cot(x\left)\right) \\ (f\circ g)\left(x\right)=-2cot\left(x\right) \end{gathered}$$

Step 3. Graph of (f∘g)(x)

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

Step 4. Composition of function 

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

role="math" localid="1646344376679" $$\begin{gathered} (g\circ f)\left(x\right)=g\left(f\right(x\left)\right) \\ (g\circ f)\left(x\right)=g(-2x) \\ (g\circ f)\left(x\right)=cot(-2x) \end{gathered}$$

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

Step 5. Graph of (g∘f)(x)

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