Question

Use
`f(x) = x^(2) +1` with domain [0, ∞] and
`g(x) = \sqrt[]{x-4}` to find each of the following.

1.
`[f^(-1)` °
`g^(-1)](x)`

2.
`[f` °
`g]^(-1) (x)`

Answer 1

1.
`(f^(-1)og^(-1)) (x) = \sqrt{x^(2)+3 }`

2.
`(f o g)^(-1) (x) = x + 3`

Explanation:

f(x) =
`x^(2)` + 1

putting f (x) = y

y =
`x^(2)` + 1

`x^(2)` = ( y-1)

x =
`(y - 1)^{(1)/(2) }`

Therefore =
`f^(-1)(x)` =
`(x-1)^{(1)/(2) }`

g (x) =
`√(x-4)`

putting g(x) = y

y =
`√(x-4)`

`y^(2)` = (x-4)

x =
`y^(2) + 4`

`g^(-1) (x) = x^(2) + 4`

(
`f^(-1)` o
`g^(-1)) x`

`(g^(-1)x -1)^{(1)/(2) } = \sqrt{x^(2) +3}`

2. (f o g)(x) = (√x-4)²+1) = x-4+1

(f o g)(x) = x-3

Let y = x - 3

x = y + 3

Or
`(f o g)^(-1)(x) = x + 3`