How to Find Domain and Range
A step-by-step tutorial on identifying domain and range from algebraic equations and graphs with worked examples and interactive plotting.
1. Definitions: What are Domain & Range?
Understanding these two core concepts is critical for algebra and calculus:
- Domain: The set of all possible input values (x-values) that do not break the function.
- Range: The set of all resulting output values (y-values) that the function actually produces.
2. The Two Golden Rules for Domain
Unless specified otherwise, assume the domain is "all real numbers," except when encountering two major mathematical red flags:
- Division by Zero: You cannot divide by zero. Any x-value that makes the denominator zero must be excluded from the domain.
- Square Roots of Negatives: In real-number math, you cannot take the square root (or any even root) of a negative number. The expression under the radical must be greater than or equal to zero (
≥ 0).
3. Worked Examples
Example A: f(x) = x²
Can you square any real number? Yes. No fractions, no square roots.
Example B: f(x) = 1 / x
Notice the variable is in the denominator. Setting x = 0 makes it divide by zero!
Example C: f(x) = √(x - 3)
The value under the root must be non-negative: x - 3 ≥ 0 ⇒ x ≥ 3.
Visualize Domain & Range on a Live Graph
Plot the functions above to inspect where the graph exists horizontally (domain) and vertically (range):
Frequently Asked Questions
What is domain in maths?
The domain is the complete set of all possible input values (usually the x-values) for which the function is defined and produces a real output.
What is range in maths?
The range is the complete set of all possible output values (usually the y-values or f(x)) that a function can produce from its domain.
How do you write domain in interval notation?
Interval notation uses brackets like [ and ] for included values, and parentheses like ( and ) for excluded values or infinity. For example, [0, ∞) means all numbers greater than or equal to 0.
What is the domain of f(x) = 1/x?
The domain is all real numbers except x = 0, because dividing by zero is undefined.
Can domain and range be all real numbers?
Yes, many functions like basic linear functions (e.g., f(x) = x) or odd-degree polynomials have both a domain and range of all real numbers.
How do you find the domain of a square root function?
Because the square root of a negative number is not a real number, set the expression inside the radical (the radicand) greater than or equal to zero (radicand >= 0) and solve for x. For example, for f(x) = √(2x - 6), set 2x - 6 >= 0 giving x >= 3, which in interval notation is [3, ∞).
How do you find the domain when a square root is in the denominator?
When a square root is in the denominator, two conditions apply: the radicand must be non-negative, and the denominator cannot equal zero. Combine these by setting the radicand strictly greater than zero (radicand > 0). For f(x) = 1/√(x - 4), set x - 4 > 0 giving x > 4, which in interval notation is (4, ∞).