Question

Let `f(x)=x^2` and `g(x)=sqrtx`, how do you find the domain and rules of `(f@g)(x)`?

Answer 1

Please see below.

Explanation:

The domain of a composition `f @ g` is the part of domain(`g`) for which `g(x)` is in domain(`f`).

In this question `g(x) = sqrtx` has domain `[0,oo)`. so the domain of the composition can't be any bigger than that.

`f(x) = x^2` has domain all real numbers. So every `g(x)` is in the domain of `f`

The domain of `f@g` is `[0,oo)`.

The rule

`(f@g)(x) = f(g(x))`

Replace `g(x)` by `sqrtx`

`(f@g)(x) = f(sqrt(x))`

For any input `(u)`, `f` gives `(u)^2`, so

`f(sqrtx) = (sqrtx)^2`

But `(sqrtx)^2 = x`

So we finish with

`(f@g)(x) = x` where `x` is in `[0.oo)`