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Range — Definition, Examples & Formula

Range

The set of y-values of a function or relation. More generally, the range is the set of values assumed by a function or relation over all permitted values of the independent variable(s).

 

Example 1: For y = x² + 1, the smallest y-value is 1, so the range is [1, ∞) or {y | y ≥ 1}.
Ellipse on xy-plane with points (-1,1), (5,1), (2,3), (2,-1) labeled. Range is [-1,3]; y-values run from -1 to 3.

 

 

See also

Domain, interval notation, set-builder notation, real numbers

Key Formula

Range of f={y∣y=f(x) for some x∈Domain}\text{Range of } f = \{ y \mid y = f(x) \text{ for some } x \in \text{Domain} \}
Where:
  • ff = The function or relation
  • xx = The independent variable (input), drawn from the domain
  • yy = The dependent variable (output), a value produced by f

Worked Example

Problem: Find the range of the function f(x) = x² − 4 where the domain is all real numbers.
Step 1: Identify the type of function. This is a quadratic that opens upward (the coefficient of x² is positive), so it has a minimum value at its vertex.
f(x)=x2−4f(x) = x^2 - 4
Step 2: Find the vertex. For f(x) = x² − 4, the vertex occurs at x = 0.
f(0)=02−4=−4f(0) = 0^2 - 4 = -4
Step 3: Determine the direction. Since the parabola opens upward, the y-value at the vertex is the minimum output. The function can produce every value from −4 upward to infinity.
Step 4: Write the range using interval notation.
Range=[−4,∞)\text{Range} = [-4, \infty)
Answer: The range is [-4, ∞), meaning y ≥ −4.

Another Example

This example involves a restricted domain that limits the range to a closed interval, unlike the first example where the range extended to infinity. It also shows that you sometimes need to find the domain before you can determine the range.

Problem: Find the range of the function g(x) = √(9 − x²).
Step 1: First, find the domain. The expression under the square root must be non-negative.
9−x2≥0  ⟹  −3≤x≤39 - x^2 \geq 0 \implies -3 \leq x \leq 3
Step 2: Find the smallest output. The square root function always returns values ≥ 0. The minimum occurs when 9 − x² = 0, which happens at x = −3 or x = 3.
g(±3)=9−9=0g(\pm 3) = \sqrt{9 - 9} = 0
Step 3: Find the largest output. The expression 9 − x² is maximized when x = 0.
g(0)=9−0=3g(0) = \sqrt{9 - 0} = 3
Step 4: Since the function is continuous on [−3, 3], it takes every value between its minimum and maximum. Write the range.
Range=[0,3]\text{Range} = [0, 3]
Answer: The range is [0, 3].

Frequently Asked Questions

What is the difference between range and domain?
The domain is the set of all allowed input values (x-values) for a function, while the range is the set of all resulting output values (y-values). Think of the domain as what you can put into the function and the range as what comes out. Every function has both a domain and a range.
How do you find the range of a function from a graph?
Look at the y-axis and identify the lowest and highest y-values the graph reaches. The range includes every y-value where the graph has at least one point. Scan the graph from bottom to top and note which y-values are covered. Use interval notation or set-builder notation to express the result.
What is the difference between range and codomain?
The codomain is the set of all possible output values that a function is defined to map into, while the range (also called the image) is the set of output values the function actually produces. The range is always a subset of the codomain. For example, f(x) = x² with codomain ℝ has range [0, ∞), which is only part of ℝ.

Range vs. Domain

RangeDomain
DefinitionThe set of all output values (y-values) of a functionThe set of all allowed input values (x-values) of a function
Which axis on a graphVertical axis (y-axis)Horizontal axis (x-axis)
How to find itDetermine all possible outputs by analyzing the function's behaviorIdentify all inputs where the function is defined (no division by zero, negatives under even roots, etc.)
Example for f(x) = √x[0, ∞)[0, ∞)
Example for f(x) = 1/x(−∞, 0) ∪ (0, ∞)(−∞, 0) ∪ (0, ∞)

Why It Matters

Finding the range is a core skill in algebra, precalculus, and calculus courses. You need it to understand what outputs a function can produce, which is essential when solving equations (if a value is not in the range, the equation has no solution). Range also appears in real-world modeling — for instance, knowing the range of a profit function tells you the maximum and minimum possible profits.

Common Mistakes

Mistake: Confusing the range with the domain, especially when both happen to be the same set (like for f(x) = x or f(x) = 1/x).
Correction: Always ask: am I looking at the inputs or the outputs? The domain concerns x-values you can plug in; the range concerns y-values that come out.
Mistake: Assuming the range of every polynomial is all real numbers (−∞, ∞).
Correction: Only odd-degree polynomials have a range of all real numbers. Even-degree polynomials, like f(x) = x², have a bounded range on one side because they have either a minimum or maximum value. Always check the function's behavior.

Related Terms