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$2x^2+1,4x^2-4x+3,2x^3+4x-x^2-2$respectively

Step-by-step solution

Given information

Given the function

$$\begin{gathered} f\left(x\right)=2x-1 \\ g\left(x\right)=x^2+2 \end{gathered}$$

Use composition and calculate

Using composition, we get

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

Use composition and calculate

Using composition, we get

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

Use composition and calculate

Using composition, we get

$$\begin{gathered} (f·g)\left(x\right)=f\left(x\right)·g\left(x\right) \\ (f·g)\left(x\right)=(2x-1)·\left(x^2+2\right) \\ (f.g)\left(x\right)=2x^3+4x-x^2-2 \end{gathered}$$