Question

In the following exercises, find

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

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

c.$(f.g)x$

$$\begin{gathered} f\left(x\right)=4x+3 \\ g\left(x\right)=2x+5 \end{gathered}$$

Short answer

The composition is$8x+23,8x+11,8x^2+26x+15$respectively

Step-by-step solution

Given information

Given the function

$$\begin{gathered} f\left(x\right)=4x+3 \\ g\left(x\right)=2x+5 \end{gathered}$$

Use composition and calculate

Using composition, we get

$$\begin{gathered} f\circ g\left(x\right)=f\left[g\right(x\left)\right] \\ f\circ g\left(x\right)=f(2x+5) \\ f\circ g\left(x\right)=4(2x+5)+3 \\ f\circ g\left(x\right)=8x+20+3 \\ f\circ g\left(x\right)=8x+23 \end{gathered}$$

Use composition and calculate

Using composition, we get

$$\begin{gathered} g\circ f\left(x\right)=g\left[f\right(x\left)\right] \\ g\circ f\left(x\right)=g(4x+3) \\ g\circ f\left(x\right)=2(4x+3)+5 \\ g\circ f\left(x\right)=8x+6+5 \\ g\circ f\left(x\right)=8x+11 \end{gathered}$$

Using composition and calculate

$$\begin{gathered} (f·g)\left(x\right)=f\left(x\right)·g\left(x\right) \\ (f·g)\left(x\right)=(4x+3)(2x+5) \\ (f·g)\left(x\right)=8x^2+20x+6x+15 \\ (f·g)\left(x\right)=8x^2+26x+15 \end{gathered}$$