Question

For functions $f\left(x\right)=3x-2$and

$g\left(x\right)=5x+1$, find

1. $(f\circ g)\left(x\right)$
   
2. $(g\circ f)\left(x\right)$
   
3. $(f·g)\left(x\right)$
   

Short answer

Part a. $(f\circ g)\left(x\right)=15x+1$

Part b. $(g\circ f)\left(x\right)=15x-9$

Part c.$(f·g)\left(x\right)=15x^2-7x-2$

Step-by-step solution

Part (a) Step 1. Find (f∘g)(x)

Using the definition of composition of functions $(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)$

Now substitute $5x+1$for $g\left(x\right)$.

$(f\circ g)\left(x\right)=f(5x+1)$

Find $f(5x+1)$where $f\left(x\right)=3x-2$.

$$\begin{gathered} (f\circ g)\left(x\right)=3(5x+1)-2 \\ (f\circ g)\left(x\right)=15x+3-2 \\ (f\circ g)\left(x\right)=15x+1 \end{gathered}$$

Part (b) Step 1. Find (g∘f)(x)

Using the definition of composition of function we have $(g\circ f)\left(x\right)=g\left(f\right(x\left)\right)$.

Now substitute $3x-2$for $f\left(x\right)$.

$(g\circ f)\left(x\right)=g(3x-2)$

Find $g(3x-2)$where $g\left(x\right)=5x+1$.

$$\begin{gathered} (g\circ f)\left(x\right)=5(3x-2)+1 \\ (g\circ f)\left(x\right)=15x-10+1 \\ (g\circ f)\left(x\right)=15x-9 \end{gathered}$$

Part (c) Step 1. Find (f·g)(x)

In this, we are multiplying the two functions. So we get

$(f·g)\left(x\right)=f\left(x\right)·g\left(x\right)$

Substitute $3x-2$for $f\left(x\right)$and $5x+1$for $g\left(x\right)$and simplify.

$$\begin{gathered} (f·g)\left(x\right)=(3x-2)(5x+1) \\ (f·g)\left(x\right)=15x^2+3x-10x-2 \\ (f·g)\left(x\right)=15x^2-7x-2 \end{gathered}$$