Question

g(x) = 5x2 - 4x
h(x) =√x - 7
Find the domain of: g(h(x))

Answer 1

\[\text{dom}(g(h(x))=\text{range}(h(x))\cap \text{dom}(g(x))\]
In other words, what are the common elements in the range of h and the domain of g?

Answer 2

Well, get the domain of h(x) first. Since x is in the radical sign and we don't want any negative numbers in there for even indices (e.g. 2), the domain of h(x) is x>=0. For g(x), it is actually the set of real numbers, since it is a polynomial. So there isn't any case for g(x).

Therefore, the only condition is for x to be >=0. So the domain of h(g(x)) is x>=0.