Question

find$(f\circ g)\left(x\right)$ and $(g\circ f)\left(x\right)$and graph each of these functions.
role="math" localid="1646345864735" $$\begin{gathered} f\left(x\right)=\frac{1}{2}x \\ g\left(x\right)=2cscx \end{gathered}$$

Short answer

If $f\left(x\right)=\frac{1}{2}x$and $g\left(x\right)=2cscx$then $(f\circ g)\left(x\right)=csc\left(x\right)$and its graph is

and $(g\circ f)\left(x\right)=2csc\left(\frac{1}{2}x\right)$and its graph is

Step-by-step solution

Step 1. Given data

The given functions are

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

Step 2. Composition of function 

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

role="math" localid="1646346006367" $$\begin{gathered} (f\circ g)\left(x\right)=f\left(g\right(x\left)\right) \\ (f\circ g)\left(x\right)=f(2csc(x\left)\right) \\ (f\circ g)\left(x\right)=\frac{1}{2}(2csc(x\left)\right) \\ (f\circ g)\left(x\right)=csc\left(x\right) \end{gathered}$$

Step 3. Graph of (f∘g)(x)

Plot the graph of $(f\circ g)\left(x\right)=csc\left(x\right)$

Step 4. Composition of function 

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

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

Step 5. Graph of (g∘f)(x)

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