Question

How do you verify that `f(x)=x^2+2, x>=0; g(x)=sqrt(x-2)` are inverses?

Answer 1

Find the inverses of the individual functions.

Explanation:

First we find the inverse of `f`:

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

To find the the inverse, we interchange x and y since the domain of a function is the co-domain (or range) of the inverse.
`f^-1: x = y^2+2`
`y^2=x-2`
`y = +-sqrt(x-2)`

Since we are told that `x>=0`,
then it means that `f^-1(x)=sqrt(x-2)=g(x)`
This implies that `g` is the inverse of `f`.

To verify that `f` is the inverse of `g` we have to repeat the process for `g`

`g(x)=sqrt(x-2)`
`g^-1: x=sqrt(y-2)`
`x^2=y-2`
`g^-1(x)=x^2-2=f(x)`
Hence we have established that `f` is an inverse of `g` and `g` is an inverse of `f`. Thus the functions are inverses of each other.