Question

In the following exercises, find

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

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

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

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

Short answer

The composition is$15x-10,15x-8,15x^2-14x+3$respectively

Step-by-step solution

Given information

Given the function

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

Use the composition and calculate

Using composition, we get

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

Using composition and calculating

Using composition, we get

$$\begin{gathered} g\circ f\left(x\right)=g\left[f\right(x\left)\right] \\ g\circ f\left(x\right)=g(3x-1) \\ g\circ f\left(x\right)=5(3x-1)-3 \\ g\circ f\left(x\right)=15x-5-3 \\ g\circ f\left(x\right)=15x-8 \end{gathered}$$

Using composition and calculate

Using composition, we get

$$\begin{gathered} (f·g)\left(x\right)=f\left(x\right)·g\left(x\right) \\ (f·g)\left(x\right)=(3x-1)·(5x-3) \\ (f·g)\left(x\right)=15x^2-9x-5x+3 \\ (f·g)\left(x\right)=15x^2-14x+3 \end{gathered}$$