Question

How do you verify if `f(x)=x^2+5; x>=0; g(x)=sqrt(x-5)` are inverse functions?

Answer 1

If `(f @ g)(x) = x` and `(g @ f)(x) = x` then they are inverse functions

Explanation:

1. See if `(f @ g)(x) = x` and `(g @ f)(x) = x`:
   `(f @ g)(x) = (sqrt(x-5))^2 + 5 = x-5+5 = x`
   `(g @ f)(x) =sqrt(x^2+5-5) = sqrt(x^2) = x`
   Therefore `f(x)` and `g(x)` are inverse functions.

2. You can also start with one of the functions, let `f(x) = y`, switch the `x` and `y` variables and solve for the `y`:
   `y = x^2 +5`
   Switch `x` and `y`: `x = y^2 + 5`
   Solve for `y`: `x-5 = y^2; y = sqrt(x-5) = g(x)`
   Therefore `f(x)` and `g(x)` are inverse functions.

3. You can graph the two functions and the `y=x` function. If the two functions are inverse functions, they will reflect each other along the `y=x` line. Make sure when you look at the graphs, you take into account any domain restrictions (e.g. `x >=0`).