Finding the domain of a function involves identifying all possible input values (often represented as 'x') that will produce a valid output (often represented as 'y'). Essentially, it's determining what values you can plug into the function without breaking any mathematical rules.
Here's a breakdown of how to find the domain:
1. Identify Potential Restrictions:
The most common restrictions that limit the domain of a function are:
- Division by Zero: A denominator cannot equal zero. If your function has a variable in the denominator, you need to find the values that make the denominator zero and exclude them from the domain.
- Square Roots (or other even roots) of Negative Numbers: You cannot take the square root (or any even root like the 4th root, 6th root, etc.) of a negative number and get a real number result. Therefore, the expression inside the square root must be greater than or equal to zero.
- Logarithms of Non-Positive Numbers: You can only take the logarithm of positive numbers. The argument of a logarithm (the expression inside the logarithm) must be greater than zero.
- Tangents, Secants, Cotangents, and Cosecants: For trigonometric functions, look for values where the functions are undefined. For example, tangent and secant are undefined when cosine is zero; cotangent and cosecant are undefined when sine is zero.
2. Analyze the Function and Set Up Inequalities (If Necessary):
Based on the restrictions you identified, set up equations or inequalities to represent the conditions that must be true for the function to be defined.
Examples:
- f(x) = 1/(x - 3): Division by zero is a concern. Set the denominator equal to zero: x - 3 = 0. Solving for x, we get x = 3. Therefore, x cannot equal 3.
- g(x) = √(x + 2): The expression inside the square root must be greater than or equal to zero. Set up the inequality: x + 2 ≥ 0. Solving for x, we get x ≥ -2.
- h(x) = ln(x - 1): The expression inside the logarithm must be greater than zero. Set up the inequality: x - 1 > 0. Solving for x, we get x > 1.
3. Solve for x:
Solve the equations or inequalities you set up in the previous step to find the values that must be excluded from the domain or the range of values that are allowed in the domain.
4. Express the Domain:
Express the domain using interval notation, set notation, or a number line.
- Interval Notation: Uses parentheses and brackets to indicate whether endpoints are included. For example,
(-∞, 3) ∪ (3, ∞)means all real numbers except 3.[-2, ∞)means all real numbers greater than or equal to -2. - Set Notation: Uses curly braces and a variable to define the set of allowed values. For example,
{x | x ≠ 3}means the set of all x such that x is not equal to 3. - Number Line: Graph the allowed values on a number line.
Examples of Expressing Domains:
- For f(x) = 1/(x - 3), the domain is
(-∞, 3) ∪ (3, ∞)or{x | x ≠ 3}. - For g(x) = √(x + 2), the domain is
[-2, ∞)or{x | x ≥ -2}. - For h(x) = ln(x - 1), the domain is
(1, ∞)or{x | x > 1}.
Special Cases:
- Polynomials: Polynomial functions (e.g., f(x) = x² + 3x - 5) have a domain of all real numbers, which can be written as
(-∞, ∞)or{x | x ∈ ℝ}. - Rational Functions: These are functions that can be written as a fraction, where the numerator and denominator are polynomials. The key is to avoid division by zero.
- Functions with Radicals: Remember to consider the index (the small number indicating the root) of the radical. Even roots have the restriction mentioned above. Odd roots (cube root, fifth root, etc.) do not have restrictions on negative numbers, so if the only restriction is an odd root, the domain is all real numbers.
In summary, to find the domain of a function, identify any potential restrictions (division by zero, even roots of negative numbers, logarithms of non-positive numbers, etc.), analyze the function, solve any resulting equations or inequalities, and express the domain appropriately.
