### Instruction: What is the domain of the function $$u(x) = \frac{1}{\sqrt x}~?$$ Express your answer in interval notation.### Response: To find the domain of a function, I need to identify the possible values of x that make the function well-defined and real-valued.
For this function, I see that there is a square root in the denominator, so I need to avoid two things: zero and negative numbers.
If x is zero, then the denominator becomes zero, which would make the function undefined, since I cannot divide by zero.
If x is negative, then the denominator becomes imaginary, which would make the function non-real, since I cannot take the square root of a negative number.
So I need to restrict x to be positive, or x > 0.
In interval notation, this means that the domain is (0, infinity), where the parenthesis indicates that 0 is not included.