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.

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 \(\color{blue}{x = 0}\)).

Algebra I for Beginners 2026 The Ultimate Step by Step Guide to Acing Algebra I

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

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

1. Look at the type of function (polynomial, rational, radical, etc.).
2. Domain: Start with all real numbers and remove values that cause division by zero or a negative under a square root.
3. Range: Find the minimum or maximum output value using the vertex (for quadratics) or algebra, then express the range in interval notation.
4. 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.

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

Answers

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

The Ultimate Algebra Bundle 2026 From Pre-Algebra to Algebra II

Download

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

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 does “all real numbers” mean in interval notation?

It is written (−∞, ∞). The parentheses (not brackets) are used because −∞ and ∞ are not actual numbers that can be reached, only directions.

Why is a square root domain restricted to non-negative values?

In the real-number system, the square root of a negative number is not a real number. To keep the output real, the radicand (expression inside the square root) must \(\color{blue}{\text{ be } \ge 0}\).

How do I write domain in set notation?

Set notation uses curly braces and a rule: for example, {x | x ≠ 5} means “all x such that x is not equal to 5.” Interval notation is often quicker, but both are correct.

Related Topics

• Function Notation
• Increasing and Decreasing Functions
• Maxima and Minima of a Function
• Graphing Rational Functions
• Composition of Functions