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$6x^2-2,12x^2-1,6x^3-2x$respectively

Step-by-step solution

Given information

Given the function

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

Use the composition and calculate

Using composition, we get

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

Use the composition and calculate

Using composition and calculate, we get

$$\begin{gathered} g\circ f\left(x\right)=g\left[f\right(x\left)\right] \\ g\circ f\left(x\right)=g\left(2x\right) \\ g\circ f\left(x\right)=3{\left(2x\right)}^2-1g\circ f\left(x\right)=12x^2-1 \end{gathered}$$

Use the composition and calculate

Using composition, we get

$$\begin{gathered} (f·g)\left(x\right)=f\left(x\right)·g\left(x\right) \\ (f·g)\left(x\right)=\left(2x\right)·\left(3x^2-1\right) \\ (f.g)\left(x\right)=2x·3x^2-2x·1 \\ (f.g)\left(x\right)=6x^3-2x \end{gathered}$$