Question

If f(x)=x62 and g(x)= sqrt x-3 find fog and its domain.

Answer 1

f(g)=(sqrt(x-3))^2=x-3 or

x=f(g)+3

fog=x+3

Answer 2

(fog)(x) = f(g(x)) = [g(x)]^2
(fog)(x) = [sqrt(x-3)]^2
note that the sqrt and ^2 will just cancel each other out:
(fog)(x) = x - 3

to find its domain,, recall that domain is the set of all possible values of x,, since we both considered f(x) and g(x) here, we find the intersection of their domains.
for f(x) = x^2 , all real numbers are possible values of x
for g(x) = sqrt(x-3) , note that numbers less than 3 cannot be substituted here because the expression will become imaginary, for example if we take x = 2
g(-2) = sqrt(2-3) = sqrt (-1) = imaginary / does not exist
thus domain of g(x) is all numbers greater than or equal to 3

the intersection of the domains is thus
all numbers greater than or equal to 3 or in symbols,
[3, +infinity)

hope this helps~ :)