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Range of A Square Root Function Calculator

Reviewed by Calculator Editorial Team

The range of a square root function is the set of all possible output values (y-values) that the function can produce. For the basic square root function f(x) = √x, the range is all non-negative real numbers. This calculator helps you determine the range of square root functions with different domains.

What is the Range of a Square Root Function?

The range of a function is the complete set of possible output values (y-values) that the function can produce for all valid input values (x-values) in its domain. For the square root function f(x) = √x, the range is all real numbers y such that y ≥ 0.

This is because the square root of any real number is always non-negative, and there is no real number whose square root is negative. The smallest value in the range is 0 (when x = 0), and the range extends infinitely upwards.

Note: The square root function is defined only for non-negative real numbers in its domain. The principal (non-negative) square root is typically used in most mathematical contexts.

How to Calculate the Range of √x

To determine the range of a square root function, follow these steps:

  1. Identify the domain of the function. For f(x) = √x, the domain is all real numbers x ≥ 0.
  2. Determine the minimum and maximum values of the function within its domain.
  3. The range will be all y-values from the minimum to the maximum output of the function.

For the basic square root function f(x) = √x:

  • The minimum value occurs at x = 0, where f(0) = 0.
  • As x increases, f(x) increases without bound.
  • Therefore, the range is [0, ∞).
Range of f(x) = √x: [0, ∞)

Worked Examples

Example 1: Basic Square Root Function

Consider the function f(x) = √x with domain x ≥ 0.

To find the range:

  1. Evaluate the function at the lower bound of the domain: f(0) = √0 = 0.
  2. As x approaches infinity, √x approaches infinity.
  3. Therefore, the range is all real numbers y ≥ 0, or [0, ∞).

Example 2: Square Root Function with Restricted Domain

Consider the function f(x) = √(x - 2) with domain x ≥ 2.

To find the range:

  1. Evaluate the function at the lower bound of the domain: f(2) = √(2 - 2) = 0.
  2. As x approaches infinity, √(x - 2) approaches infinity.
  3. Therefore, the range is all real numbers y ≥ 0, or [0, ∞).

Example 3: Square Root Function with Upper Bound

Consider the function f(x) = √(9 - x²) with domain -3 ≤ x ≤ 3.

To find the range:

  1. Evaluate the function at critical points: f(-3) = √(9 - 9) = 0, f(0) = √(9 - 0) = 3, f(3) = √(9 - 9) = 0.
  2. The maximum value of the function is 3 (at x = 0).
  3. The minimum value is 0 (at x = -3 and x = 3).
  4. Therefore, the range is [0, 3].

Frequently Asked Questions

What is the range of the square root function?
The range of the basic square root function f(x) = √x is all non-negative real numbers, or [0, ∞).
Can the range of a square root function be negative?
No, the principal square root function always yields non-negative results. The range cannot include negative numbers.
How does the domain affect the range of a square root function?
The domain determines the x-values for which the function is defined. The range is then all y-values produced by the function for those x-values. For example, if the domain is restricted to x ≥ 4, the range will start from √4 = 2.
What is the range of √(x²)?
The range of √(x²) is [0, ∞) because squaring any real number and then taking the square root returns the absolute value, which is always non-negative.
Can a square root function have a finite range?
Yes, if the domain is restricted to a finite interval where the function attains a maximum value. For example, f(x) = √(9 - x²) on the domain [-3, 3] has a range of [0, 3].