The domain of a mathematical function is the complete set of possible input values (often denoted as $x$) for which the function is defined. In other words, it's the set of all values that can be plugged into the function without resulting in undefined behavior, such as division by zero or taking the square root of a negative number (in the case of real functions).

For example:

• For the function $f(x) = \frac{1}{x}$, the domain is all real numbers except $x = 0$, because division by zero is undefined.
• For the function $g(x) = \sqrt{x}$, the domain is all non-negative real numbers ($x \geq 0$), because square roots of negative numbers are not real.

To find the domain of a function, one typically looks for restrictions such as:

1. Division by zero.
2. Even roots of negative numbers (if working with real numbers).
3. Logarithms of non-positive numbers.

Would you like a more detailed explanation of domain for specific functions or examples?

Here are 5 related questions to expand on this topic:

1. What is the domain of the function $f(x) = \frac{1}{x-3}$?
2. How do you determine the domain of a square root function like $f(x) = \sqrt{4 - x^2}$?
3. Can functions have different domains depending on the context (e.g., real vs complex numbers)?
4. What is the domain of a logarithmic function, such as $f(x) = \ln(x)$?
5. How does the domain of a polynomial function differ from a rational function?

Tip: When determining the domain of a function, always start by checking for values that would make any part of the expression undefined (like division by zero or negative square roots).