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Domain of a Function: Definition, How to Find It + Examples

Domain

The set of values of the independent variable(s) for which a function or relation is defined. Typically, this is the set of x-values that give rise to real y-values.

Note: Usually domain means domain of definition, but sometimes domain refers to a restricted domain.

See also

Range, interval notation, set-builder notation

Key Formula

Domain of f={x∈R∣f(x) is defined}\text{Domain of } f = \{ x \in \mathbb{R} \mid f(x) \text{ is defined} \}
Where:
  • xx = The independent variable (input) of the function
  • f(x)f(x) = The function rule applied to x
  • R\mathbb{R} = The set of all real numbers

Worked Example

Problem: Find the domain of f(x) = 1 / (x − 3).
Step 1: Identify operations that restrict the domain. This function has a fraction, so the denominator cannot equal zero.
x−3≠0x - 3 \neq 0
Step 2: Solve for the excluded value by setting the denominator equal to zero.
x−3=0  ⟹  x=3x - 3 = 0 \implies x = 3
Step 3: The domain is all real numbers except x = 3. Write this in set-builder notation.
{x∈R∣x≠3}\{ x \in \mathbb{R} \mid x \neq 3 \}
Step 4: Equivalently, express the domain in interval notation.
(−∞,3)∪(3,∞)(-\infty, 3) \cup (3, \infty)
Answer: The domain is all real numbers except 3, written as (−∞, 3) ∪ (3, ∞).

Another Example

This example involves a square root restriction (radicand ≥ 0), unlike the first example which involved a denominator restriction (denominator ≠ 0). It shows that different operations create different types of domain restrictions.

Problem: Find the domain of g(x) = √(2x − 6).
Step 1: Identify the restricting operation. A square root requires its radicand (the expression inside) to be greater than or equal to zero.
2x−6≥02x - 6 \geq 0
Step 2: Solve the inequality for x.
2x≥6  ⟹  x≥32x \geq 6 \implies x \geq 3
Step 3: Write the domain in interval notation. Since x = 3 is included (the square root of 0 is defined), use a square bracket.
[3,∞)[3, \infty)
Answer: The domain is x ≥ 3, or [3, ∞) in interval notation.

Frequently Asked Questions

How do you find the domain of a function?
Start with all real numbers and then remove any values that cause undefined operations. The most common restrictions are: denominators that equal zero (division by zero is undefined), negative numbers under even-index radicals (square roots, fourth roots, etc.), and negative or zero arguments inside logarithms. Whatever x-values remain after removing these form the domain.
What is the difference between domain and range?
The domain is the set of all valid input values (x-values) you can plug into a function. The range is the set of all output values (y-values) the function actually produces. Think of the domain as what goes in, and the range as what comes out.
What is the domain of a polynomial function?
Every polynomial function (such as f(x) = x², f(x) = 3x³ − 2x + 1, etc.) has a domain of all real numbers, written as (−∞, ∞). Polynomials involve only addition, subtraction, multiplication, and non-negative integer exponents, so no input value causes an undefined result.

Domain vs. Range

DomainRange
DefinitionSet of all valid input values (x-values)Set of all resulting output values (y-values)
What it answers"What can I put in?""What can come out?"
How to find itLook for values that cause undefined operations (division by zero, negative square roots, etc.)Analyze the function's behavior, graph, or solve for x in terms of y
On a graphThe horizontal extent (left-right spread) of the graphThe vertical extent (up-down spread) of the graph
Example for f(x) = x²(−∞, ∞)[0, ∞)

Why It Matters

Domain appears throughout algebra, precalculus, and calculus whenever you define or analyze a function. You need the domain to graph functions accurately, solve equations involving radicals or rational expressions, and set up real-world models where inputs must make physical sense (for example, a length cannot be negative). In calculus, knowing the domain is essential for determining where a function is continuous and differentiable.

Common Mistakes

Mistake: Only excluding values where the denominator is zero but forgetting about square root restrictions (or vice versa).
Correction: Check every operation in the function. For a function like f(x) = √(x) / (x − 4), you need both x ≥ 0 (square root) and x ≠ 4 (denominator), giving the domain [0, 4) ∪ (4, ∞).
Mistake: Confusing domain with range — stating the y-values when asked for the domain.
Correction: Domain always refers to the input (independent variable, usually x). Range refers to the output (dependent variable, usually y). If you're asked for the domain, focus on which x-values are allowed.

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