Question

I NEED HELP WITH QUESTION PLEASE!!
What is the inverse of f(x)
(x) = (x-4)² for x ≥ 4?

Answer 1

4th choice
`\bold{f^(-1)(x) = √(x) + 4}`

Explanation:

Definition of the inverse of a function

A function g is the inverse of a function f if whenever y=f(x) then x=g(y). In other words, applying f and then g is the same thing as doing nothing. We can write this in terms of the composition of f and g as g(f(x))=x. The domain of f becomes the range of g and the range of f becomes the domain of g

To solve for the inverse of the function
`f(x) =\left(x-4\right)^2`

Let
`y=\left(x-4\right)^2`

`\mathrm{Replace}\:x\:\mathrm{with}\:y`
`\text{ and replace }\:y\:\mathrm{with}\:x`

`x=\left(y-4\right)^2`

Switch sides

`\left(y-4\right)^2=x`

Take square roots on both sides

`y-4=\pm√(x)`

Add 4 on both sides to solve for y

`y = \pm√(x) + 4`

We have two solutions

`y=√(x)+4,\:y=-√(x)+4`

To determine which one of these to be chosen not that in the given choices we can eliminate the first two since x cannot be negative

The third choice can also be eliminated since

`-√(x) + 4` is a decreasing function for
`x \ge 0`

So the last answer choice is correct and the inverse of
`f(x) = (x-4)^2`

is given by
`f^(-1)(x) = √(x) + 4`

Answer:4th choice
`\bold{f^(-1)(x) = √(x) + 4}`

Note

Domain of (x-4)² is [4, ∞) since x ≥ 4 and (x-4)² cannot be negative

Range of (x-4)² is [0, ∞)

Domain of
`√(x)\:+\:4` is [0, ∞)

Range of
`√(x)\:+\:4` is [4, ∞)

so indeed the domain of (x-4)² has become the range of
`√(x)\:+\:4` and the range of (x-4)² has become the domain of
`√(x)\:+\:4`