How to Find Domain and Range of Radical Functions? (+FREE Worksheet!)

Finding the domain and range of radical functions is an important skill in Algebra 1 and pre-calculus. Because you cannot take the square root of a negative number (in the real number system), the domain of a square-root function is restricted to values that keep the radicand non-negative. The range then follows from the behavior of the function on that domain. This guide explains both concepts clearly with four worked examples, two video lessons, and practice problems.

What Are Domain and Range?

The domain of a function is the set of all input values (x-values) for which the function is defined. The range is the set of all output values (y-values) that the function can produce.

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For a radical function \(\color{blue}{f(x) = \sqrt{(\text{ expression })}}\), the domain is all x-values that make the expression inside the radical greater than or equal to zero.

How to Find the Domain of a Radical Function

For Square Root Functions

Set the \(\color{blue}{\text{ radicand } \ge 0}\) and solve for x. The domain is the solution set written in interval notation.

• \(\color{blue}{f(x) = \sqrt{(x – 3)}}\): require \(\color{blue}{x – 3 \ge 0}\) → \(\color{blue}{x \ge 3}\). Domain: \(\color{blue}{[3, \infty )}\).
• \(\color{blue}{f(x) = \sqrt{(2x + 6)}}\): require \(\color{blue}{2x + 6 \ge 0}\) → \(\color{blue}{x \ge -3}\). Domain: \(\color{blue}{[-3, \infty )}\).
• \(\color{blue}{f(x) = \sqrt{(-x + 4)}}\): require \(\color{blue}{-x + 4 \ge 0}\) → \(\color{blue}{x \le 4}\). Domain: \(\color{blue}{(-\infty , 4]}\).

For Cube Root Functions

Cube roots are defined for all real numbers (positive, negative, and zero). The domain of any cube-root function is all real numbers: \(\color{blue}{(-\infty , \infty )}\).

How to Find the Range of a Radical Function

For a basic square root function \(\color{blue}{f(x) = a\sqrt{(x – h)} + k}\):

• If \(\color{blue}{a > 0}\), the function increases from its minimum value. The minimum output is \(\color{blue}{k}\) (when \(\color{blue}{x = h}\)). Range: \(\color{blue}{[k, \infty )}\).
• If \(\color{blue}{a < 0}\), the function decreases from its maximum value. The maximum output is \(\color{blue}{k}\). Range: \(\color{blue}{(-\infty , k]}\).

Evaluate \(\color{blue}{f}\) at the left (or right) endpoint of the domain to find the starting output value.

Step-by-Step Summary

1. Identify the radicand (the expression under the radical sign).
2. Set the \(\color{blue}{\text{ radicand } \ge 0}\) (for a square root) and solve for x — this gives the domain.
3. Identify the vertex / starting point of the function (where the radicand equals 0).
4. Determine whether the function increases or decreases (sign of the leading coefficient).
5. State the range based on the starting output value and direction.

Watch: Domain of a Radical Function (Video Lesson)

Khan Academy shows how to set the \(\color{blue}{\text{ radicand } \ge 0}\) and convert the answer to interval notation:

Domain and Range of Radical Functions — Worked Examples

Example 1: Find the domain of \(\color{blue}{f(x) = \sqrt{(x – 3)}}\).

Require: \(\color{blue}{x – 3 \ge 0}\) → \(\color{blue}{x \ge 3}\).
Domain: \(\color{blue}{[3, \infty )}\). Range: \(\color{blue}{[0, \infty )}\) (output starts at 0 when \(\color{blue}{x = 3}\)).

Example 2: Find the domain of \(\color{blue}{f(x) = \sqrt{(2x + 6)}}\).

Require: \(\color{blue}{2x + 6 \ge 0}\) → \(\color{blue}{2x \ge -6}\) → \(\color{blue}{x \ge -3}\).
Domain: \(\color{blue}{[-3, \infty )}\). Range: \(\color{blue}{[0, \infty )}\).

Example 3: Find the domain and range of \(\color{blue}{f(x) = \sqrt{(-x + 4)}}\).

Require: \(\color{blue}{-x + 4 \ge 0}\) → \(\color{blue}{x \le 4}\).
Domain: \(\color{blue}{(-\infty , 4]}\). As x decreases, \(\color{blue}{-x + 4}\) increases, so the range is \(\color{blue}{[0, \infty )}\).

Example 4: Find the domain and range of \(\color{blue}{f(x) = -2\sqrt{(x + 1)} + 3}\).

Require: \(\color{blue}{x + 1 \ge 0}\) → \(\color{blue}{x \ge -1}\). Domain: \(\color{blue}{[-1, \infty )}\).
At \(\color{blue}{x = -1}\): \(\color{blue}{f(-1) = -2(0) + 3 = 3}\). As x increases, \(\color{blue}{\sqrt{(x+1)}}\) increases, so \(\color{blue}{-2\sqrt{(x+1)}}\) decreases toward \(\color{blue}{-\infty}\).
Range: \(\color{blue}{(-\infty , 3]}\).

Graphing and Domain & Range Video

The Organic Chemistry Tutor shows how to graph radical functions and read domain and range from the graph:

Exercises for Domain and Range of Radical Functions

Find the domain (and range, where noted) of each function.

1. \(\color{blue}{f(x) = \sqrt{(x + 5)}}\)
2. \(\color{blue}{f(x) = \sqrt{(3 – x)}}\)
3. \(\color{blue}{f(x) = \sqrt{(4x – 8)}}\)
4. \(\color{blue}{f(x) = \sqrt{(x^{2} – 9)}}\)
5. \(\color{blue}{f(x) = 2\sqrt{(x – 1)} + 4}\) — find domain AND range

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Answers

1. Domain: \(\color{blue}{[-5, \infty )}\)
2. Domain: \(\color{blue}{(-\infty , 3]}\)  (require \(\color{blue}{3 – x \ge 0}\))
3. Domain: \(\color{blue}{[2, \infty )}\)  (require \(\color{blue}{4x – 8 \ge 0 \rightarrow x \ge 2}\))
4. Domain: \(\color{blue}{x \le -3}\) or \(\color{blue}{x \ge 3}\), i.e., \(\color{blue}{(-\infty , -3] \cup [3, \infty )}\)  (require \(\color{blue}{x^{2} \ge 9}\))
5. Domain: \(\color{blue}{[1, \infty )}\); Range: \(\color{blue}{[4, \infty )}\)  (minimum value is \(\color{blue}{f(1) = 4}\))

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Free Domain and Range of Radical Functions Worksheet

Ready to practice on your own? Download our free Domain and Range of Radical Functions 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 Domain and Range of Radical Functions before a quiz or test.

Download Domain and Range Worksheet

Frequently Asked Questions

Why is the domain of a square root restricted?

In the real number system, you cannot take the square root of a negative number — the result would be imaginary. Therefore, only x-values that make the \(\color{blue}{\text{ radicand } \ge 0}\) are allowed in the domain.

How do you write the domain in interval notation?

Use square brackets \(\color{blue}{[}\) and \(\color{blue}{]}\) when the endpoint is included (the inequality uses ≥ or ≤), and parentheses \(\color{blue}{(}\) and \(\color{blue}{)}\) when the endpoint is excluded or the interval extends to infinity. For example, \(\color{blue}{x \ge 3}\) is written \(\color{blue}{[3, \infty )}\).

What is the range of f(x) = √x?

The basic square root function \(\color{blue}{f(x) = \sqrt{x}}\) has domain \(\color{blue}{[0, \infty )}\) and range \(\color{blue}{[0, \infty )}\). The output is never negative because √x always returns the principal (non-negative) square root.

Related Topics

• Radical Equations
• Simplifying Radical Expressions
• Rationalizing Radical Expressions
• Adding and Subtracting Radical Expressions
• Multiplying Radical Expressions