How to Find Domain and Range of a Function: Every Method for 2026

Domain and range are the bookends of every function. Domain is the set of allowable inputs; range is the set of possible outputs. The topic shows up on every algebra test, every pre-calc quiz, and most calculus questions about function behavior. Most students get half of the points and lose the rest because they remember the definitions but not the restrictions.

This guide covers every method for finding domain and range, the four restriction rules that drive most problems, and the interval notation conventions you need on test day.

Definitions

• Domain: all x-values you can plug into the function.
• Range: all y-values the function can produce.

A useful question to ask yourself: “What values of x am I not allowed to use?” The answer is your restriction list. Everything else is the domain.

The Four Restriction Rules

Almost every domain restriction in Algebra 2 traces back to one of these:

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1. Denominators cannot be zero. For 1/(x − 3), x ≠ 3.
2. Square roots (and any even root) cannot be negative inside. For √(x − 4), x ≥ 4.
3. Logarithms require positive arguments. For log(x + 2), x > −2.
4. Real-world context restricts to non-negative or integer inputs. Time, height, and count cannot be negative.

Memorize the four. Almost no other restrictions appear in high school.

Interval Notation

Square brackets include the endpoint; parentheses exclude it. Infinity is always paired with a parenthesis because infinity is not a number you can reach.

Domain From an Equation: Step by Step

1. Identify the function type (polynomial, rational, radical, log, etc.).
2. Apply the relevant restriction rules.
3. Solve any inequality or equality that arises.
4. Write the answer in interval notation.

Example 1: Polynomial

f(x) = 3x² − 5x + 1.

No denominators, no roots, no logs. Domain: (−∞, ∞).

Polynomials have no restrictions. Their domain is always all real numbers.

Example 2: Rational Function

f(x) = (x + 2) / (x² − 9).

Denominator zero when x² − 9 = 0 → x = ±3. Exclude both.

Domain: (−∞, −3) ∪ (−3, 3) ∪ (3, ∞).

Example 3: Square Root

f(x) = √(2x − 6).

2x − 6 ≥ 0 → x ≥ 3.

Domain: [3, ∞).

Example 4: Square Root in a Denominator

f(x) = 1 / √(x − 1).

Two restrictions:
– Inside root is non-negative: x − 1 ≥ 0 → x ≥ 1.
– Denominator is not zero: x − 1 ≠ 0 → x ≠ 1.

Combined: x > 1.

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Domain: (1, ∞).

Example 5: Logarithm

f(x) = log(x − 5).

Argument must be positive: x − 5 > 0 → x > 5.

Domain: (5, ∞).

Example 6: Combined Restrictions

f(x) = √(x + 4) / (x − 2).

Two restrictions:
– x + 4 ≥ 0 → x ≥ −4.
– x − 2 ≠ 0 → x ≠ 2.

Combined: x ≥ −4 and x ≠ 2.

Domain: [−4, 2) ∪ (2, ∞).

Domain From a Graph

Project every point of the graph onto the x-axis. The shadow on the x-axis is the domain.

• A graph that stretches in both directions infinitely: domain is (−∞, ∞).
• A graph with a vertical asymptote at x = 3: domain excludes 3.
• A graph that starts at x = 2 and goes right: domain is [2, ∞) if it includes the endpoint, (2, ∞) if it does not.

Pay attention to filled vs. open circles. Filled means included; open means excluded.

Range: The Harder Half

Range from an equation is harder than domain because you have to think about what y can do.

Polynomials

• Odd-degree polynomials (linear, cubic): range is (−∞, ∞).
• Even-degree polynomials (quadratic, quartic): range depends on the vertex and the leading coefficient. For y = ax² + bx + c with a > 0, the range is [vertex y, ∞).

Rational Functions

Range is everything except the horizontal asymptote, plus possibly the asymptote if the function crosses it.

f(x) = 1/x. Horizontal asymptote y = 0; the function never equals 0. Range: (−∞, 0) ∪ (0, ∞).

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Radicals

f(x) = √(x − 4). Output is non-negative. Range: [0, ∞).

Absolute Value

y = |x − 3| + 2. Absolute value is non-negative; minimum value is at x = 3. Range: [2, ∞).

Range From a Graph

Project every point onto the y-axis. The shadow is the range.

This is the easiest way to find range, period. If the equation is messy, sketch a quick graph and read it off.

Function Notation and Domain

f(x) means “f of x,” the output when the input is x.

f(3) is the output when x = 3. To find f(3), plug 3 into the formula. f(3) is defined only if 3 is in the domain.

When a question asks “for what values of x is f defined?” it is asking for the domain.

Common Mistakes

1. Forgetting the denominator-zero restriction. Always check.
2. Treating cube roots like square roots. Cube roots (and odd roots) accept negative inputs. ∛(x − 2) has domain (−∞, ∞).
3. Confusing ≥ and >. Square roots use ≥ (zero is allowed). Logs use > (zero is not allowed).
4. Writing infinity with a square bracket. Always (∞) or (−∞).
5. Mixing up which axis is which. Domain reads off the x-axis; range reads off the y-axis.

A Quick Cheat Sheet

Frequently Asked Questions

Is “domain” the same as “input”?
The domain is the entire set of allowable inputs. Any single value from the domain is an input.

Does every function have a range that includes all real numbers?
No. Only some functions (like linear, cubic, and tangent) have range equal to all real numbers. Most have a restricted range.

How do I find range without a graph?
Use the function type. Quadratics have a vertex-based range. Radicals are non-negative. Exponentials are positive. Logs are all reals. Cubics are all reals.

What is the difference between domain and codomain?
Codomain is the set we are mapping to (often all reals); range is the set we actually hit. In high school, “range” usually means “image.”

Are domain and range on the SAT?
Domain shows up regularly, especially for rational and radical functions. Range less often. Both appear constantly on Algebra 2 finals.

Closing Thought

Domain is a checklist of four restriction rules. Range is whatever the function actually outputs, found from the graph or from the function type. Master the restriction rules, learn interval notation cold, and you have ten easy points per quiz waiting.

For more practice, browse our Algebra 2 worksheets and our full Math Topics library. When you are ready for a structured workbook, our Algebra 2 collection covers domain and range in depth.

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