Question

100 Points!!!! Please Show Steps!

1. Given f(x)=x^2 - 2 and g(x) =4x + 1 Find

a. f(g(x))
b. g(f(x))

Answer 1

`f[g(x)]=16x^2+8x-1`

`g[f(x)]=4x^2-7`

Explanation:

Given functions:

`f(x)=x^2 - 2`

`g(x) =4x + 1`

`f[g(x)]` means to substitute g(x) in place of x in f(x):

`\begin{aligned}\implies f[g(x)] & =[g(x)]^2-2\\ & =(4x+1)^2-2\\ & = (4x+1)(4x+1)-2\\ & = 16x^2+8x+1-2\\ & = 16x^2+8x-1\end{aligned}`

`g[f(x)]` means to substitute f(x) in place of x in g(x):

`\begin{aligned}\implies g[f(x)] & =4[f(x)]+1\\ & = 4(x^2-2)+1\\ & = 4x^2-8+1\\ & = 4x^2-7\end{aligned}`

Answer 2

• f(x) = x² - 2

• g(x) = 4x + 1

Go from right to left in such function questions

a. f(g(x))

⇒ f(4x + 1)

⇒ (4x + 1)² - 2

⇒ 16x² + 8x + 1 - 2

⇒ 16x² + 8x - 1

b. g(f(x))

⇒ g(x² - 2)

⇒ 4(x² - 2) + 1

⇒ 4x² - 7