The range of a function is the set of all output values (usually y-values) the function actually produces from its domain. For f(x) = x2, every output is 0 or more, so the range is y ≥ 0. On a graph, the range is how far the graph reaches down and up.
The range is every output the graph reaches. This parabola never goes above its vertex, so its range is y ≤ 4.
Formal definition and notation
For a functionf with domain D, the range is the set {f(x) ∣ x ∈ D}: every value y for which some input x in the domain gives f(x) = y. The range is a subset of the codomain, the set the outputs are allowed to come from; for real functions in algebra it is a set of real numbers.
set-builder form: all outputs for inputs in the domain
Key facts
The range depends on the domain: f(x) = 2x on 0 ≤ x ≤ 3 has range 0 ≤ y ≤ 6.
Even powers and absolute values cannot be negative: |x| and x2 both have range y ≥ 0.
Adding a constant shifts the range: x2 − 5 has range y ≥ −5.
Different inputs can share an output, so the range can be smaller than the domain: {(1, 7), (2, 7), (3, 7)} has range {7}.
Square root functions start at a boundary: √x + 2 has range y ≥ 2.
How to find the range
How to find the range
Find the lowest and highest outputs, then ask which values in between are reached.
graph
read bottom to top: the lowest and highest points of the graph give the ends
parabola
the vertex gives the minimum (opens up) or maximum (opens down) output
ordered pairs
list the second coordinates, each once
solve for x
a value y is in the range if f(x) = y has a solution in the domain
A linear function that is not constant has range all real numbers; a constant function f(x) = 5 has range {5}.
Try it: draw a range
Interactive
The shaded ray is the range of the parabola in the picture: y ≤ 4. Drag the endpoint or flip the arrow to show the range of an upward parabola, such as y ≥ −2.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Range of a Function examples, step by step
Example 1: Range of ordered pairs
Warm-up
Find the range of {(−2, 6), (0, 1), (3, 6), (5, −4)}.
The outputs are the second coordinates: 6, 1, 6, −4.
List each value once, in order.
Answer: The range is {-4, 1, 6}.
Example 2: Range of a parabola
Core
Find the range of f(x) = x2 + 6x + 5.
The parabola opens up (a = 1 > 0), so its vertex gives the lowest output.
Vertex x-value: −62(1) = −3.
Lowest output: f(−3) = 9 − 18 + 5 = −4.
Every output from −4 upward is reached.
Answer: The range is y ≥ -4, or [−4, ∞).
Example 3: Range on a limited domain
Test-style
A tank holds V(t) = 500 − 20t liters of water t minutes after it starts draining, for 0 ≤ t ≤ 25. What is the range of V?
V is linear and decreasing, so the ends of the domain give the ends of the range.
At t = 0: V = 500. At t = 25: V = 500 − 500 = 0.
Every amount between is reached as the water drains.
Answer: The range is 0 ≤ V ≤ 500 liters.
Common mistakes
✗ Mistakef(x) = x2 − 3 has range all real numbers, like its domain.
✓ Insteadx2 ≥ 0, so x2 − 3 ≥ −3: the range is y ≥ −3.
A square is never negative, so the outputs have a floor. Find the vertex or the smallest possible output.
✗ MistakeFor y = −(x − 1)2 + 4, the range is x ≤ 1.
✓ InsteadThe range uses y: y ≤ 4, from the vertex (1, 4).
The x-coordinate of the vertex tells where the maximum happens; the y-coordinate tells what the maximum output is.
✗ MistakeThe range of a data set, max minus min, is the range of a function: {(1, 3), (2, 9)} has range 6.
✓ InsteadAs a function, its range is the set {3, 9}.
In statistics, range is one number. For a function, the range is a set of outputs.
Domain is the set of inputs (x); range is the set of outputs (y). For y = √x, both are [0, ∞), but for y = x2 + 1 the domain is all reals and the range is y ≥ 1.
The statistical range is a single number, the largest value minus the smallest: for 3, 8, 11 it is 8. The range of a function is a set of outputs.
Range of a Functionvs.Codomain
The codomain is the set outputs are allowed to come from, often all real numbers. The range is the part actually reached: for x2, only y ≥ 0.
Quick check: 3 questions
Question 1 of 3
What is the range of f(x) = |x| + 2?
Show answer and explanation
Answer: B.|x| ≥ 0, so |x| + 2 ≥ 2, and every value 2 or more is reached. y ≥ 0 forgot the shift up, x ≥ −2 describes inputs, and all real numbers ignores that absolute value cannot be negative.
Question 2 of 3
A parabola opens down and has vertex (−2, 7). What is its range?
Show answer and explanation
Answer: D. Opening down makes the vertex the highest point, so the outputs go from 7 down: y ≤ 7. y ≥ 7 is for a parabola opening up, y ≤ −2 used the x-coordinate, and x ≤ 7 mixed up the variables.
Question 3 of 3
g(x) = 3x − 1 has domain {0, 1, 2}. What is its range?
Show answer and explanation
Answer: A. Evaluate each input: g(0) = −1, g(1) = 2, g(2) = 5. {0, 1, 2} is the domain, all real numbers ignores the small domain, and {0, 3, 6} forgot the −1.
Range of a Function: frequently asked questions
How do you find the range of a function from a graph?
Read the graph from bottom to top. Find the lowest point and the highest point the graph reaches, and check whether each end is included (closed dot), excluded (open dot) or continues forever (arrow). Every y-value between them that the graph touches is in the range. For a parabola, the vertex is one end.
Is the range the same as the codomain?
No. The codomain is the set the outputs are allowed to come from, usually all real numbers in algebra. The range is the set of outputs that actually happen. For f(x) = x2, the codomain can be all real numbers, but the range is only the numbers 0 and above, because no real number squared is negative.
What is the range of a linear function?
All real numbers, as long as the slope is not 0 and the domain is all real numbers. A line like y = −2x + 7 keeps rising in one direction and falling in the other, so it hits every y-value. A horizontal line y = 7 has range {7}, and a limited domain gives a limited range.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-IF.A.1
Cite this entry
Effortless Math Team. “Range of a Function.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/range-of-a-function/
Flashcard
TermRange of a Functionoutputs: y
DefinitionAll the outputs a function actually produces. f(x) = x2 − 1 never goes below −1, so its range is y ≥ −1.
DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2SubsetA subset is a set whose elements all belong to another set. If every element of A is also in B, then A is a subset of B, written A ⊆ B. Every set is a subset of itself, the empty set is a subset of every set, and a set with n elements has 2n subsets.Probability · Algebra 1, StatisticsReal NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1InequalityAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Algebra · Grade 6, Grade 7, Grade 8Interval NotationInterval notation is a short way to write a set of numbers between two endpoints, using brackets and parentheses. A bracket, [ or ], means the endpoint is included; a parenthesis, ( or ), means it is not. So −2 ≤ x < 3 is written [−2, 3), and x > 4 is written (4, ∞).Symbols and Notation · Algebra 1, Algebra 2, PrecalculusPowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometrySquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3DataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2RangeThe range of a data set is the difference between the greatest and least values: maximum minus minimum. For 2, 5, 9, the range is 9 − 2 = 7. Range is a simple measure of spread: it tells how far apart the extreme values are, though one outlier can make it misleading about the rest of the data.Statistics · Grade 5, Grade 6, Grade 7SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1OutputAn output is the value a function or rule produces from a given input. You put a number in, apply the rule, and the result that comes out is the output. For y = 2x, the input 3 gives the output 6. In an input-output table the output sits in the second column.Functions · Grade 6, Grade 7, Grade 8Dependent VariableA dependent variable is the output value that changes based on the input. It responds to changes in the independent variable: in y = 2x + 1, y is the dependent variable because its value depends on x. On a graph it goes on the vertical axis, because its value depends on what the input was.Functions · Grade 6, Grade 7, Grade 8Domain and RangeDomain and range describe a function: the domain is the set of all inputs it accepts, and the range is the set of all outputs it produces. For the points (1, 4) and (2, 5), the domain is {1, 2} and the range is {4, 5}. Restrictions usually come from division by zero or square roots of negatives.Functions · Grade 8, Algebra 1, Algebra 2