Question

For functions $f\left(x\right)=x^2-9$and

$g\left(x\right)=2x+5$, find

1. $(f\circ g)(-2)$
   
2. $(g\circ f)(-3)$
   
3. $(f\circ f)\left(4\right)$
   

Short answer

Part a. $(f\circ g)(-2)=-8$

Part b. $(g\circ f)(-3)=5$

Part c.$(f\circ f)\left(4\right)=40$

Step-by-step solution

Part (a) Step 1. Find (f∘g)(-2)

Using the definition of composition of functions we have $(f\circ g)(-2)=f\left(g\right(-2\left)\right)$.

Find $g(-2)$where $g\left(x\right)=2x+5$.

$$\begin{gathered} (f\circ g)(-2)=f(2·(-2)+5) \\ (f\circ g)(-2)=f(-4+5) \\ (f\circ g)(-2)=f\left(1\right) \end{gathered}$$

Find $f\left(1\right)$where $f\left(x\right)=x^2-9$

$$\begin{gathered} (f\circ g)(-2)=1^2-9 \\ (f\circ g)(-2)=1-9 \\ (f\circ g)(-2)=-8 \end{gathered}$$

Part (b) Step 1. Find (g∘f)(-3)

Using the definition of composition of functions we have $(g\circ f)(-3)=g\left(f\right(-3\left)\right)$

Find $f(-3)$where $f\left(x\right)=x^2-9$.

$$\begin{gathered} (g\circ f)(-3)=g({(-3)}^2-9) \\ (g\circ f)(-3)=g(9-9) \\ (g\circ f)(-3)=g\left(0\right) \end{gathered}$$

Find $g\left(0\right)$where $g\left(x\right)=2x+5$

$$\begin{gathered} (g\circ f)(-3)=2·0+5 \\ (g\circ f)(-3)=0+5 \\ (g\circ f)(-3)=5 \end{gathered}$$

Part (c) Step 1. Find (f∘f)(4)

Using the definition of composition of functions we have $(f\circ f)\left(4\right)=f\left(f\right(4\left)\right)$.

Find $f\left(4\right)$where $f\left(x\right)=x^2-9$

$$\begin{gathered} (f\circ f)\left(4\right)=f(4^2-9) \\ (f\circ f)\left(4\right)=f(16-9) \\ (f\circ f)\left(4\right)=f\left(7\right) \end{gathered}$$

Now find $f\left(7\right)$where $f\left(x\right)=x^2-9$.

$$\begin{gathered} (f\circ f)\left(4\right)=7^2-9 \\ (f\circ f)\left(4\right)=49-9 \\ (f\circ f)\left(4\right)=40 \end{gathered}$$