Question

In Problems 2 – 4, for the given functions f and g find:

(a) (f o g)(2)

(b) (g o f) (-2)

(c) (f o f)(4)

(d) ( g o g) (-1)

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

Short answer

$$\begin{gathered} \left(a\right)(fog)\left(2\right)=e^4 \\ \left(b\right)(gof)(-2)=3e^{-2}-2 \\ \left(c\right)(fof)\left(4\right)=e^{e^4} \\ \left(d\right)(gog)(-1)=-17 \end{gathered}$$

Step-by-step solution

Part(a) Step 1: Given Information

We are given with two functions

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

Part(a) : Step 2: Composition of functions

$$\begin{gathered} f\left(x\right)=e^x \\ g\left(x\right)=3x-2 \\ (fog)\left(2\right)=f\left(g\right(2\left)\right) \\ g\left(2\right)=3\left(2\right)-2=4 \\ f\left(4\right)=e^4 \\ (fog)\left(2\right)=e^4 \end{gathered}$$

Part(b) : Step 1: composition of functions

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

Part(c): Step 1: composition of functions

$$\begin{gathered} f\left(x\right)=e^x \\ g\left(x\right)=3x-2 \\ (fof)\left(4\right)=f\left(f\right(4\left)\right) \\ f\left(4\right)=e^4 \\ f\left(e^4\right)=e^{e^4} \\ (fof)\left(4\right)=e^{e^4} \end{gathered}$$

Part (d) : Step 1: composition of funcitons

$$\begin{gathered} f\left(x\right)=e^x \\ g\left(x\right)=3x-2 \\ (gog)(-1)=g\left(g\right(-1\left) \\ g\right(-1)=3(-1)-2=-5 \\ g(-5)=3(-5)-2=-17 \\ (gog)(-1)=-17 \end{gathered}$$