Composite functions question

State the domain of f ° g(x) and g ° f(x) if f(x) = √x and g(x) = 2x - 4

I'm just starting to work with composite functions, I have the solution to this question, but I cannot grasp how it works, can someone please give me some pointers? Here's the answer:

f ° g(x) = f(g(x))

=f(2x-4)

=√(2x-4)***

f ° g(x) is defined as long as 2x - 4 ≥ 0 ; 2x ≥4 ; x ≥ 4/2 ; x ≥ 2

∴ the domain of f ° g(x) is {x | x ≥ 2, x ∈ R}

g ° f(x) = g(f(x))

=g(√x)

=2(√x) - 4***

g ° f(x) is defined as long as x ≥ 0

∴ the domain of g° f(x) is {x | x ≥ 0, x ∈ R}

***These marked parts are where I am becoming confused, I can't figure out what is being done to 2x-4 to make √(2x-4), or what is being done to √x to make 2(√x) - 4.

Re: Composite functions question

Quote:

Originally Posted by

**Lethargic** State the domain of f ° g(x) and g ° f(x) if f(x) = √x and g(x) = 2x - 4

Assuming that the functions are mapping on the same set then:

$\displaystyle \text{Dom}(f\circ g)=\text{Dom}(f)\cap\text{Dom}( g)$