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$6x^2-9x,18x^2-9x,6x^3-9x^2$respectively

Step-by-step solution

Given information

Given the function

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

Use the composition and calculate

Using composition, we get

$$\begin{gathered} f\circ g\left(x\right)=f\left[g\right(x\left)\right] \\ f\circ g\left(x\right)=f(2x^2-3x) \\ f\circ g\left(x\right)=3(2x^2-3x) \\ f\circ g\left(x\right)=6x^2-9x \end{gathered}$$

Use composition and calculate

Using composition, we get

$$\begin{gathered} g\circ f\left(x\right)=g\left[f\right(x\left)\right] \\ g\circ f\left(x\right)=g\left(3x\right) \\ g\circ f\left(x\right)=2{\left(3x\right)}^2-3\left(3x\right) \\ g\circ f\left(x\right)=18x^2-9x \end{gathered}$$

Use composition and calculate

Using composition, we get

$$\begin{gathered} (f.g)\left(x\right)=f\left(x\right).g\left(x\right) \\ (f.g)\left(x\right)=\left(3x\right)·\left(2x^2-3x\right) \\ (f.g)\left(x\right)==3x·2x^2-3x·3x \\ (f.g)\left(x\right)=6x^3-9x^2 \end{gathered}$$