How to Find the Domain of a Square Root Function

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In mathematics, the domain of a function tells you for which values of ​x​ the function is valid. This means that any value within that domain will work in the function, while any value that falls outside of the domain will not. Some functions (such as linear functions) have domains that include all possible values of ​x​. Others (such as equations where ​x​ appears within the denominator) exclude certain values of ​x​ to avoid dividing by zero. Square root functions have more restricted domains than some other functions, since the value within the square root (known as the radicand) has to be a positive number for the result to be "real."

TL;DR (Too Long; Didn't Read)

The domain of a square root function is all values of ​x​ that result in a radicand that is equal to or greater than zero.

Square Root Functions

A square root function is a function that contains a radical, which is more commonly called a square root. If you aren't sure what this looks like,

f(x) = \sqrt{x}

is considered a basic square root function. In this case, ​x​ cannot be a negative number; all radicals must be equal to or greater than zero for the result to be real. If you can include "imaginary" numbers (with ​i​ defined as the square root of −1) then things get more complicated, but in most cases you only need to consider real numbers.

This doesn't mean that all square root functions are as simple as the square root of a single number. More complex square root functions may have calculations within the radical, calculations that modify the radical's result or even a radical as part of a larger function (such as appearing in the numerator or denominator of an equation). Examples of these more complex functions look like

f(x) = 2\sqrt{x + 3} \text{ or } g(x) = \sqrt{x - 4}

Domains of Square Root Functions

To calculate the domain of a square root function, solve the inequality ​x​ ≥ 0 with ​x​ replaced by the radicand. Using one of the examples above, you can find the domain of

f(x) = 2\sqrt{x + 3}

by setting the radicand (​x​ + 3) equal to ​x​ in the inequality. This gives you the inequality of

x + 3 ≥ 0

which you can solve by subtracting 3 by both sides. This gives you a solution of x ≥ −3, meaning that your domain is all values of ​x​ greater than or equal to −3. You can also write this as [ −3, ∞), with the bracket on the left showing that −3 is a specific limit while the parenthesis on the right shows that ∞ is not. Since the radicand cannot be negative, you only have to calculate for positive or zero values.

Range of Square Root Functions

A concept related to the domain of a function is its range. While a function's domain is all of the values of ​x​ that are valid within the function, its range is all of the values of ​y​ in which the function is valid. This means that the range of a function equals all of the valid outputs of that function. You can calculate this by setting ​y​ equal to the function itself, and then solving to find any values that are not valid.

For square root functions, this means that the range of the function is all values produced when ​x​ results in a radicand that is equal to or greater than zero. Calculate the domain of your square root function, and then input the value of your domain into the function to determine the range. If your function is

f(x) = \sqrt{x - 2}

and you calculate the domain as all values of ​x​ greater than or equal to 2, then any valid value you put into

y = \sqrt{x - 2}

will give you a result that is greater than or equal to zero. Therefore your range is ​y​ ≥ 0 or [0, ∞).


About the Author

Holding a BS in computer science and several years of experience building, repairing and maintaining computers and electronics, Jack Gerard has had a love of science and mathematics for years. When not working on writing projects as part of his 15+ year career, he also works as a programmer writing gaming and accessibility software.