Q. 47 - 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.
$$\begin{gathered} f\left(x\right)=-2x \\ g\left(x\right)=cot x \end{gathered}$$

Step-by-Step Solution

Answer

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

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

Step 1. Given data

The given functions are

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

Step 2. Composition of function

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

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

Step 3. Graph of ( f ∘ g ) ( x )

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

Step 4. Composition of function

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

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

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

Step 5. Graph of ( g ∘ f ) ( x )

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