Q. 48 - 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.
$$\begin{gathered} f\left(x\right)=\frac{1}{2}x \\ g\left(x\right)=2 csc x \end{gathered}$$

Step-by-Step Solution

Answer

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

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

Step 1. Given data

The given functions are

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

Step 2. Composition of function

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

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

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

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

Step 5. Graph of ( g ∘ f ) ( x )

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