The domain of a function is the set of values for which it is defined. When the function involves a square root, the square root is not defined for negative arguments, so the domain cannot include any values that make the square root be of a negative number.
Similarly, a rational function will be undefined where its denominator is zero.
1) h(x) is defined for 2x+10 > 0, or x > -5. The marked choice is appropriate.
2) The product of the two functions will be
[tex](f\cdot g)(x)=\dfrac{\sqrt{x+3}}{x}\cdot\dfrac{\sqrt{x+3}}{2x}=\dfrac{\sqrt{x+3}\sqrt{x+3}}{(x)(2x)}\\\\=\dfrac{x+3}{2x^2}\qquad\text{3rd choice}[/tex]
3) The domain of f(x) is x≠0. The domain of g(x) is 3x-9>0, or x > 3. The latter includes the former, so the domain is ...
... (3, ∞)
