How to Find Domain and Range of a Function?

Finding the domain and range of a function tells you exactly which inputs a function will accept and which outputs it can produce. The domain is the set of all valid x-values, while the range is the set of all resulting y-values. Identifying domain and range is a foundational skill in Algebra 1 that you will rely on throughout every math course that follows.

Tutor-style math help

Find Domain and Range of a Function: what to notice and how to work it

Functions skill

A function is a rule that gives each input exactly one output. Function notation, tables, graphs, and equations are different ways to show the same input-output relationship.

What to notice first

Domain is the set of allowed inputs; range is the set of possible outputs. Read them from left-to-right and bottom-to-top on a graph.

Common student mistake

Do not read \(f(4)\) as multiplication. It means the output of f when the input is 4.

Key formulas and cues

\(\text{domain}=\text{allowed }x\text{-values}\)

\(\text{range}=\text{possible }y\text{-values}\)

\(\sqrt{x-a}\Rightarrow x\ge a\)

\(\frac{1}{x-a}\Rightarrow x\ne a\)

A reliable path

Identify the inputFind the x-value, expression, or inner function being used.
Apply the ruleSubstitute with parentheses so signs and powers stay clear.
Interpret the outputState the value, point, interval, domain, range, or inverse relationship.

Worked examples

Evaluate a function

Example: \(f(x)=4x-3\), find \(f(2)\)

Replace x with 2.
Compute 4(2) – 3.
Simplify.

Answer: \(5\)

Compose functions

Example: \(f(x)=x+1\), \(g(x)=2x\), find \(f(g(3))\)

Find g(3) = 6.
Use that as the input for f.
f(6) = 7.

Answer: \(7\)

Open pop-up practice

Try one before moving on

Try: If \(h(x)=2x^2\), find \(h(-3)\).

Answer: \(18\). Use parentheses: \(2(-3)^2=18\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

What Are Domain and Range?

Domain: All input values (x-values) for which the function is defined — i.e., produces a real-number output.
Range: All output values (y-values) that the function actually produces for inputs in the domain.

Written in interval notation: the domain of \(\color{blue}{f(x) = x}\)\(\color{blue}{^{2} + 1}\) is (−∞, ∞) and the range is [1, ∞) because the smallest output is 1 (when \(\color{blue}{x = 0}\)).

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How to Find the Domain of a Function

Polynomial functions

No restrictions. Domain is always all real numbers: (−∞, ∞).

Example: \(\color{blue}{f(x) = 3x}\)² − \(\color{blue}{2x + 5}\) has domain (−∞, ∞).

Functions with fractions (rational functions)

Exclude any x-value that makes the denominator equal to zero.

Example: \(\color{blue}{f(x) = \frac{1}{(x – 5)}}\). Set \(\color{blue}{\text{ denominator } = 0}\): \(\color{blue}{x – 5 = 0}\), so \(\color{blue}{x = 5}\).
Domain: all real numbers except 5, written (−∞, 5) ∪ (5, ∞).

Functions with square roots (radical functions)

The expression under the radical must be greater than or equal to zero.

Example: f(x) = √(\(\color{blue}{x – 2}\)). Set \(\color{blue}{x – 2 \ge 0}\): \(\color{blue}{x \ge 2}\).
Domain: [2, ∞).

How to Find the Range of a Function

Method 1 — Algebraic analysis

Identify the minimum or maximum output value. For \(\color{blue}{f(x) = x}\)\(\color{blue}{^{2} + 1}\), the smallest x² can be is 0 (at \(\color{blue}{x = 0}\)), so the smallest output is \(\color{blue}{0 + 1 = 1}\). Range: [1, ∞).

Method 2 — Graph inspection

Look at the graph and determine the lowest and highest y-values. Use interval notation to record what the graph covers vertically.

Step-by-Step Summary

Look at the type of function (polynomial, rational, radical, etc.).
Domain: Start with all real numbers and remove values that cause division by zero or a negative under a square root.
Range: Find the minimum or maximum output value using the vertex (for quadratics) or algebra, then express the range in interval notation.
Write answers in interval notation or set notation.

Watch: Domain and Range Given a Formula (Video Lesson)

Khan Academy demonstrates how to find the domain and range of a function given a formula with several worked examples:

Domain and Range – Worked Examples

Example 1: Find the domain of \(\color{blue}{f(x) = x}\)\(\color{blue}{^{2} + 1}\).

Polynomial: no restrictions. Domain: (−∞, ∞).
Range: \(\color{blue}{x^{2} \ge 0}\) for all x, so \(\color{blue}{f(x) \ge 1}\). Range: [1, ∞).

Example 2: Find the domain of \(\color{blue}{f(x) = \frac{1}{(x – 5)}}\).

\(\color{blue}{\text{ Denominator } = 0}\) when \(\color{blue}{x = 5}\) (excluded). Domain: (−∞, 5) ∪ (5, ∞).
Range: f(x) can be any value except 0. Range: (−∞, 0) ∪ (0, ∞).

Example 3: Find the domain of f(x) = √(\(\color{blue}{x – 2}\)).

Require \(\color{blue}{x – 2 \ge 0}\) → \(\color{blue}{x \ge 2}\). Domain: [2, ∞).
Range: √\(\color{blue}{(x – 2) \ge 0}\). Range: [0, ∞).

Example 4: Find domain and range of f(x) = −\(\color{blue}{x^{2} + 4}\).

Polynomial: Domain: (−∞, ∞).
−\(\color{blue}{x^{2} \le 0}\), so f(x) = −x² + \(\color{blue}{4 \le 4}\). Maximum is 4 at \(\color{blue}{x = 0}\). Range: (−∞, 4].

More Practice: Domain and Range of a Function (Video Lesson)

Khan Academy provides a second lesson focusing on reading domain and range from graphs and formula-based functions:

Exercises for Domain and Range

Find the domain and range of each function.

\(\color{blue}{f(x) = 2x + 7}\)
\(\color{blue}{f(x) = x}\)\(\color{blue}{^{2} – 3}\)
\(\color{blue}{f(x) = \frac{1}{(x + 4)}}\)
f(x) = √(\(\color{blue}{3x – 6}\))
f(x) = −\(\color{blue}{2x^{2} + 8}\)

Answers

Domain: (−∞, ∞); Range: (−∞, ∞)
Domain: (−∞, ∞); Range: [−3, ∞) (minimum at \(\color{blue}{x = 0}\))
Domain: (−∞, −4) ∪ (−4, ∞); Range: (−∞, 0) ∪ (0, ∞)
\(\color{blue}{3x – 6 \ge 0}\) → \(\color{blue}{x \ge 2}\). Domain: [2, ∞); Range: [0, ∞)
Domain: (−∞, ∞); Range: (−∞, 8] (maximum 8 at \(\color{blue}{x = 0}\))

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Free Finding Domain and Range Worksheet

Ready to practice on your own? Download our free Finding Domain and Range worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master Finding Domain and Range before a quiz or test.

Download Domain and Range Worksheet

Frequently Asked Questions

What is domain and range for a function?

The domain is every input (x-value) the function can accept, and the range is every output (y-value) it can actually produce. For f(x) = 2x + 1, any real number can go in and any real number can come out, so both are all real numbers. For f(x) = x², the domain is all real numbers but the range is y ≥ 0.

How do you find domain and range of a function without graphing?

For the domain, look for values that break the rule: division by zero and square roots of negative numbers. For the range, think about what outputs are possible. For f(x) = √(x − 3), the inside must be at least 0, so the domain is x ≥ 3. A square root never gives a negative result, so the range is y ≥ 0.

How do you find the domain and range of a quadratic function?

The domain of any quadratic is all real numbers. The range depends on the vertex and the direction the parabola opens. For f(x) = 9 − x², the vertex is (0, 9) and the parabola opens down, so the range is y ≤ 9. For f(x) = (x − 1)² + 4, it opens up from the vertex (1, 4), so the range is y ≥ 4.

How do you find the domain and range of a rational function?

Set the denominator equal to zero and exclude those x-values from the domain. For f(x) = 1/(x − 2), x cannot be 2, so the domain is all real numbers except 2. The fraction 1/(x − 2) can never equal 0, because its numerator is never 0, so the range is all real numbers except 0. The graph has asymptotes at x = 2 and y = 0.

How do you find domain and range from a graph?

For the domain, scan the graph from left to right and note the smallest and largest x-values it covers. For the range, scan from bottom to top for the y-values. Closed dots mean the endpoint is included, open dots mean it is not, and arrows mean the graph keeps going. A graph from (−2, 1) to (3, 6) with closed dots has domain −2 ≤ x ≤ 3 and range 1 ≤ y ≤ 6.

How do you find the domain and range of an exponential function?

An exponential function such as y = 2ˣ accepts every real number, so its domain is all real numbers. Its outputs are always positive, so the range is y > 0. A vertical shift moves the range: y = 2ˣ − 3 has range y > −3, because the horizontal asymptote moves down to y = −3.

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