What is how to find the domain of a function?

Here's a guide to finding the domain of a function:

The domain of a function is the set of all possible input values (often represented by 'x') for which the function is defined and produces a real number output. Finding the domain involves identifying any values that would cause the function to be undefined. Here are some common scenarios to consider:

1. Fractions: If the function has a fraction, the denominator cannot be zero. Set the denominator equal to zero and solve for x. These values are excluded from the domain.

   • Related concept: Division by Zero

2. Square Roots (and other even roots): The expression inside the square root (or any even root, like a fourth root) must be greater than or equal to zero. Set the expression inside the root ≥ 0 and solve for x.

   • Related concept: Radical Functions

3. Logarithms: The argument of a logarithm must be strictly greater than zero. Set the argument > 0 and solve for x.

   • Related concept: Logarithmic Functions

4. Rational Exponents: If you have a rational exponent like x^m/n, consider the following:

   • If n is even, the base (x) must be greater than or equal to 0 (same as even roots).
   • If n is odd, there are typically no restrictions unless there's another issue like division by zero involved.

5. Functions with Restricted Domains by Definition: Some functions, especially in applied contexts, might have domains restricted based on the problem's context (e.g., you can't have a negative number of items).

Expressing the Domain:

Once you've identified the values that are not in the domain, you can express the domain using:

• Interval Notation: For example, (-∞, 2) ∪ (2, ∞) means all real numbers except 2.
• Set-Builder Notation: For example, {x | x ≠ 2} means the set of all x such that x is not equal to 2.

General Strategy:

1. Identify any potential restrictions (fractions, roots, logarithms, etc.).
2. Determine the values that make the function undefined.
3. Exclude those values from the set of all real numbers.
4. Express the remaining set using interval or set-builder notation.