# Composition of functions domains

Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)?

I've been having trouble with this and this answer would make me fully understand this concept.

Thanks to everyone!

mathman
It depends on the range of g. If the range of g is contained within the domain of f, then the domain of g is the domain of f(g(x)). If the range of g is not contained, then the part of the range outside is cutoff, inducing a cutoff in the useful domain of of g, i.e. that which ends up as the domain of f(g(x))

HallsofIvy
Homework Helper
Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)?

I've been having trouble with this and this answer would make me fully understand this concept.

Thanks to everyone!
No, it is not always true. In fact it is seldom true. What is true is that the domain of f(g(x)) is that subset of the domain of g such that g(x) is in the domain of f(x).

Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)?
The domain is simply the set of values your function can accept as input.

The function f o g has the same domain as g which we can pretty safely assume to be all the real numbers.

The domain of f might be more narrow, but f o g has the same domain as g.