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The domain is the set of possible values for x. When finding domains you start by assuming that x can be any number. Then you look for reasons to restrict the domain. Reasons to restrict a domain include:
- Denominators cannot allowed to become zero
- Arguments of logarithms cannot be allowed to be zero or negative
- Radicands of even-numbered roots cannot be allowed to become negative.
You have no denominators or even-numbered roots. But you do have logarithms. We must make sure the arguments stay positive.
For h(x) the argument of the logarithm is x^2. x^2 is positive for all numbers except one: zero. So the domain of h(x) is all numbers except zero.
For g(x) the argument of the logarithm is x. To keep the argument positive, we must keep x positive. So the domain for g(x) is