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:

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.

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, ∞).

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

What is the easiest way to find domain and range? For domain, start with all real numbers and remove what breaks the function: values that make a denominator zero, make an even root negative, or make a log argument zero or negative. For range, sketch or picture the graph and read its lowest and highest y-values. For f(x) = √(x − 4), the domain is x ≥ 4 and the range is y ≥ 0.

How do you find domain and range of a graph? Read the domain from left to right along the x-axis and the range from bottom to top along the y-axis. Closed dots include an endpoint, open dots exclude it, and arrows mean the graph continues forever. If a graph starts at a closed dot at (−2, 1) and rises with an arrow to the right, the domain is [−2, ∞) and the range is [1, ∞).

How do you find domain and range from an equation? Identify the function type, apply the restriction rules, and solve any inequality that results. For f(x) = 1/(x + 5), the denominator cannot be zero, so x ≠ −5, giving a domain of (−∞, −5) ∪ (−5, ∞). Since the fraction can never equal zero, the range is all real numbers except 0. Polynomials like 2x³ − x have a domain of all real numbers.

How do you write domain and range in interval notation? Use square brackets for included endpoints and parentheses for excluded ones, and always use parentheses with infinity. So x ≥ 3 is [3, ∞), −2 < x ≤ 5 is (−2, 5], and all real numbers except 4 is (−∞, 4) ∪ (4, ∞). The union symbol ∪ joins separate pieces when a value or gap is removed from the domain.

How do you find domain and range of a quadratic function? The domain of every quadratic is all real numbers, (−∞, ∞). For the range, find the vertex. If the parabola opens up (a > 0), the range is y ≥ the vertex’s y-value; if it opens down, y ≤ that value. For f(x) = x² − 6x + 5, the vertex is at x = 6 ÷ 2 = 3 and f(3) = −4, so the range is [−4, ∞).

How do you find domain and range of a rational function? Set the denominator equal to zero and exclude those x-values from the domain. For the range, look for a horizontal asymptote and any y-values the function cannot reach. For f(x) = 3/(x − 2), the domain is all real numbers except 2, and since the horizontal asymptote is y = 0 and the function never equals 0, the range is all real numbers except 0.

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.

Before finding the domain and range from a graph, confirm the graph is a function with the vertical line test: how to tell if a graph is a function.