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

Name: How to Find Domain and Range of Radical Functions? (+FREE Worksheet!)
Uploaded: 2021-03-15T22:01:35+00:00
Description: 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

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.

Try it

Radical Simplifier

Open the full calculator →

Type the problems from this lesson and check each step.

Quick answer: For an even root like √, set the expression under the radical ≥ 0 and solve to get the domain. The range starts at the vertical shift k and goes up if the coefficient is positive, down if it is negative. Example: y = √(x − 3) + 2 has domain x ≥ 3 and range y ≥ 2. Odd roots like ∛ allow all real numbers.

Tutor-style math help

Find Domain and Range of Radical Functions: what to notice and how to work it

Radicals skill

Radicals are roots. Simplifying or solving with radicals is mostly about perfect powers, domain restrictions, and checking for extraneous answers.

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 split a radical across addition. \(\sqrt{a+b}\) is not usually \(\sqrt a+\sqrt b\).

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

Find perfect powersBreak the radicand into a perfect power times a leftover factor.
Watch the domainEven roots need nonnegative radicands in real-number problems.
Check solutionsIf you squared both sides, test answers in the original equation.

Worked examples

Simplify a radical

Example: \(\sqrt{72}\)

72 = 36 times 2.
The square root of 36 is 6.
Leave the leftover 2 inside.

Answer: \(6\sqrt2\)

Find a radical domain

Example: \(y=\sqrt{x-4}\)

The radicand is x – 4.
Require x – 4 >= 0.
Solve the inequality.

Answer: \(x\ge4\)

Open pop-up practice

Try one before moving on

Try: Simplify \(\sqrt{98}\).

Answer: \(7\sqrt2\).

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?

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.

Algebra I for Beginners 2027 The Ultimate Step by Step Guide to Acing Algebra I

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

Identify the radicand (the expression under the radical sign).
Set the \(\color{blue}{\text{ radicand } \ge 0}\) (for a square root) and solve for x. This gives the domain.
Identify the vertex / starting point of the function (where the radicand equals 0).
Determine whether the function increases or decreases (sign of the leading coefficient).
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.

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

Answers

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

The Ultimate Middle School Math Bundle Grades 6-8

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

Even index vs odd index: how the root changes the answer

The index of the radical decides whether you need to restrict x at all.

Function	Index	Domain	Range
y = √(x − 3) + 2	2 (even)	x ≥ 3	y ≥ 2
y = −2√(x + 4) + 1	2 (even)	x ≥ −4	y ≤ 1
y = ⁴√(5 − x)	4 (even)	x ≤ 5	y ≥ 0
y = ∛(x + 1)	3 (odd)	all real numbers	all real numbers
y = ∛x − 6	3 (odd)	all real numbers	all real numbers

Even index (2, 4, 6…): the radicand cannot be negative, so solve radicand ≥ 0. In y = ⁴√(5 − x), 5 − x ≥ 0 gives x ≤ 5. Odd index (3, 5…): you can take the cube root of a negative number, for example ∛(−8) = −2, so x is not restricted and the graph covers every y-value.

How to find the range of a square root function in 3 steps

Write the function as y = a√(bx + c) + k.
Find the starting point. The radical part equals 0 at the edge of the domain, so the graph starts at y = k.
Check the sign of a. If a > 0 the graph rises, so the range is y ≥ k. If a < 0 the graph falls, so the range is y ≤ k.

Example: y = −2√(x + 4) + 1. The domain is x ≥ −4. At x = −4 the radical is 0, so y = 1. Because a = −2 is negative, the range is y ≤ 1. A quick check: at x = 0, y = −2(2) + 1 = −3, which is below 1.

Watch out for a radical in a denominator. In y = 1/√(x − 2) the radicand must be strictly positive, so the domain is x > 2 and the range is y > 0.

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.

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

For an even root, set the radicand greater than or equal to zero and solve; that is the domain. For the range, find the y-value where the graph starts (the vertical shift k) and see whether the coefficient in front of the radical is positive or negative. Odd roots have all real numbers for both.

What is the domain of √x?

The domain of y = √x is x ≥ 0, written [0, ∞) in interval notation, because the square root of a negative number is not a real number. Its range is also y ≥ 0. Shifting the graph changes these, for example y = √(x + 2) has domain x ≥ −2.

How do you find the range of a radical function?

Find the starting y-value, which is k in y = a√(x − h) + k. Then look at a. If a is positive the range is y ≥ k, and if a is negative the range is y ≤ k. For odd roots such as cube roots, the range is all real numbers.

Do cube root functions have a restricted domain?

No. A cube root can be taken of any real number, including negatives, since ∛(−27) = −3. So a function like y = ∛(x − 4) + 1 has domain and range of all real numbers. Only even roots, such as square roots and fourth roots, restrict the domain.

What happens if the radical is in the denominator?

Then the radicand must be strictly greater than zero, because dividing by zero is undefined. For y = 1/√(x − 2), you need x − 2 > 0, so the domain is x > 2. The output is always positive, so the range is y > 0.

Related to This Article

Domain and Range Radical Functions

What people say about "Finding the Domain and Range of a Radical Function"?

No one replied yet.