# Domain of composite function?

Let functions f and g be defined by their graphs, a ray in both cases.

h(x) = f[g(x)]

What is the domain of the composite function h?

This was not actually my question. Someone else asked it early this year. I was trying to understand the disappointing responses. Two people (now three) argued that the domain of h was the intersection of domains of f and g. One other person said that the composite function was not possible.

For starters, there is no g(-3), so of course there can be no f[g(-3)].

I will keep checking for a while. It is not a trick question. Surely someone will get it.

### 1 Answer

- Demiurge42Lv 75 months ago
If the function composition, f(g(x)), exists, then the domain of f(g(x)) is ALWAYS the domain of g. For the function composition to exist the range of g must be a subset of the domain of f.

From your graph, the range of g is [1, -∞) and the domain of f is [-3, ∞).

[1, -∞) is not a subset of [-3, ∞) so the function composition doesn't exist.