Question

What is the inverse function of `f(x)=sqrt(8-x)`?

Answer 1

The inverse of `f(x) = sqrt(8-x)` is
`color(white)("XXXX")` `g(x)=8-x^2`

Explanation:

Rename `f(x)` as `y`
Then, using the original equation, solve for `x`
`color(white)("XXXX")` `y = sqrt(8-x)`

`color(white)("XXXX")` `y^2= 8-x`

`color(white)("XXXX")` `y^2-8 = -x`

`color(white)("XXXX")` `x = 8-y^2`

Rename `x` as `g(y)`
`color(white)("XXXX")` `g(y) = 8-y^2`

...or using the more common variable `x` as the function variable:
`color(white)("XXXX")` `g(x) = 8-x^2`

Answer 2

`f^-1(y) = 8-y^2` with domain `[0, oo)` and range `(-oo,8]`

Explanation:

Let `y = f(x) = sqrt(8-x)`

The domain of `f` is `(-oo,8]` and its range is `[0,oo)`

Then `y^2 = 8-x`

Add `x-y^2` to both sides to get:

`x = 8 - y^2`

So `f^-1(y) = 8-y^2` with domain `[0,oo)` and range `(-oo,8]`