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$24x+1,24x-19,24x^2-14x-5$respectievly

Step-by-step solution

Given information

Given the function

$$\begin{gathered} f\left(x\right)=6x-5 \\ g\left(x\right)=4x+1 \end{gathered}$$

Use the composition and calculate

Using the composition, we get

$$\begin{gathered} f\circ g\left(x\right)=f\left[g\right(x\left)\right] \\ f\circ g\left(x\right)=f(4x+1) \\ f\circ g\left(x\right)=6(4x+1)-5 \\ f\circ g\left(x\right)=24x+6-5 \\ f\circ g\left(x\right)=24x+1 \end{gathered}$$

Using the composition and calculate

Using the composition, we get

$$\begin{gathered} g\circ f\left(x\right)=g\left[f\right(x\left)\right] \\ g\circ f\left(x\right)=g(6x-5) \\ g\circ f\left(x\right)=4(6x-5)+1 \\ g\circ f\left(x\right)=24x-20+1 \\ g\circ f\left(x\right)=24x-19 \end{gathered}$$

Use the composition and calculate

Using composition, we get

$$\begin{gathered} (f·g)\left(x\right)=f\left(x\right)·g\left(x\right) \\ (f·g)\left(x\right)=(6x-5)·(4x+1) \\ (f.g)\left(x\right)=24x^2+6x-20x-5 \\ (f.g)\left(x\right)=24x^2-14x-5 \end{gathered}$$