Question

For the functions, f(x) = 6x + 1 and

g(x) = 8x − 3, find:

$$\begin{gathered} \left(a\right)(f\circ g)\left(x\right) \\ \left(b\right)(g\circ f)\left(x\right) \\ \left(c\right)(f·g)\left(x\right) \end{gathered}$$

Short answer

$$\begin{gathered} \left(a\right)(f\circ g)\left(x\right)=48x-17 \\ \left(b\right)(g\circ f)\left(x\right)=48x+5 \\ \left(c\right)(f·g)\left(x\right)=48x^2-10x-3 \end{gathered}$$

Step-by-step solution

Part (a) Step 1. Given information

Given function

$$\begin{gathered} f\left(x\right)=6x+1 \\ g\left(x\right)=8x-3 \end{gathered}$$

Part (a) Step 2. Calculation

$$\begin{gathered} f\left(x\right)=6x+1 \\ g\left(x\right)=8x-3 \\ Now,substituteg\left(x\right)=8x-3inf\left(g\right(x\left)\right) \\ f\left(g\right(x\left)\right)=f(8x-3) \\ Evaluatingf(8x-3)replacexby8x-3infunctionf\left(x\right)=6x+1 \\ f(8x-3)=6(8x-3)+1 \\ =48x-18+1 \\ =48x-17 \\ (f\circ g)\left(x\right)=48x-17 \end{gathered}$$

Part (b) Step 2. Given information

$$\begin{gathered} f\left(x\right)=6x+1 \\ g\left(x\right)=8x-3 \end{gathered}$$

Part (b) Step 2. Calculation

$$\begin{gathered} (g\circ f)\left(x\right)=g\left(f\right(x\left)\right) \\ Nowsubstitutef\left(x\right)=6x+1ing\left(f\right(x\left)\right) \\ g\left(f\right(x\left)\right)=g(6x+1) \\ Toevaluateg(6x+1)replacexby6x+1function \\ g\left(x\right)=8x-3 \\ g(6x+1)=8(6x+1)-3 \\ =48x+8-3 \\ =48+5 \\ (g\circ f)\left(x\right)=48x+5 \end{gathered}$$

Part (c) Step 1. Given information

$$\begin{gathered} f\left(x\right)=6x+1 \\ g\left(x\right)=8x-3 \end{gathered}$$

Part (c) Step 2. Calculation

$$\begin{gathered} Toevaluate(f·g)\left(x\right), \\ (f·g)\left(x\right)=f\left(x\right).g\left(x\right) \\ =(6x+1)·(8x-3) \\ =48x^2-18x+8x-3 \\ =48x^2-10x-3 \end{gathered}$$