Question

In the following exercises, find the values described.

For functions ,$f\left(x\right)=2x^2+3$and

$g\left(x\right)=5x-1$find

a.$f\circ g(-2)$,

b.$g\circ f(-3)$

c.$f\circ f(-1)$

Short answer

The value is$305,104,53$respectively

Step-by-step solution

Given information

Given the function

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

Use composition and calculate

Using composition, we get

$$\begin{gathered} f\circ g\left(x\right)=f\left[g\right(x\left)\right] \\ f\circ g\left(x\right)=f(5x-1) \\ f\circ g\left(x\right)=2{(5x-1)}^2+3 \\ f\circ g\left(x\right)=2(25x^2-10x+1)+3 \\ f\circ g\left(x\right)=50x^2-20x+2+3 \\ f\circ g\left(x\right)=50x^2-20x+5 \\ f\circ g(-2)=50{(-2)}^2-20(-2)+5 \\ f\circ g(-2)=200+100+5 \\ f\circ g(-2)=305 \end{gathered}$$

Use the composition and calculate

Using composition, we get

$$\begin{gathered} g\circ f\left(x\right)=g\left[f\right(x\left)\right] \\ g\circ f\left(x\right)=g(2x^2+3) \\ g\circ f\left(x\right)=5(2x^2+3)-1 \\ g\circ f\left(x\right)=10x^2+15-1 \\ g\circ f\left(x\right)=10x^2+14 \\ g\circ f(-3)=10{(-3)}^2+14 \\ g\circ f(-3)=104 \end{gathered}$$

Use composition and calculate

Using composition, we get

$$\begin{gathered} f\circ f\left(x\right)=f\left[f\right(x\left)\right] \\ f\circ f\left(x\right)=f(2x^2+3) \\ f\circ f\left(x\right)=2{(2x^2+3)}^2+3 \\ f\circ f\left(x\right)=2(4x^4+12x^2+9)+3 \\ f\circ f\left(x\right)=8x^4+24x^2+18+3 \\ f\circ f\left(x\right)=8x^4+24x^2+21 \\ f\circ f(-1)=8{(-1)}^4+24{(-1)}^2+21 \\ f\circ f(-1)=8+24+21 \\ f\circ f(-1)=53 \end{gathered}$$