Question

In the following exercises, find

a.$f\circ g\left(x\right)$

b.$g\circ f\left(x\right)$

c.$(f.g)\left(x\right)$

Short answer

The composition is$6x-1,6x+17,6x^2+13x-28$respectively

Step-by-step solution

Given information

Given the function

$$\begin{gathered} f\left(x\right)=2x+7 \\ g\left(x\right)=3x-4 \end{gathered}$$

Use composition and calculate

Using composition, we get

$$\begin{gathered} f\circ g\left(x\right)=f\left[g\right(x\left)\right] \\ f\circ g\left(x\right)=f(3x-4) \\ f\circ g\left(x\right)=2(3x-4)+7 \\ f\circ g\left(x\right)=6x-8+7 \\ f\circ g\left(x\right)=6x-1 \end{gathered}$$

Use composition and calculate

Using composition, we get

$$\begin{gathered} g\circ f\left(x\right)=g\left[f\right(x\left)\right] \\ g\circ f\left(x\right)=g(2x+7) \\ g\circ f\left(x\right)=3(2x+7)-4 \\ g\circ f\left(x\right)=6x+21-4 \\ g\circ f\left(x\right)=6x+17 \end{gathered}$$

Use composition and calculate

Using composition, we get

$$\begin{gathered} (f·g)\left(x\right)=f\left(x\right)·g\left(x\right) \\ (f·g)\left(x\right)=(2x+7)·(3x-4) \\ (f.g)\left(x\right)=6x^2-8x+21x-28 \\ (f.g)\left(x\right)=6x^2+13x-28 \end{gathered}$$