Domain 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.
The domain is every input the function uses; the range is every output it reaches. On a graph, the domain is the shadow on the x-axis and the range is the shadow on the y-axis.
Formal definition and notation
For a functionf, the domain is the set of all x-values for which f(x) is defined (or that make sense in the situation), and the range is the set of all values f(x) takes for x in the domain. For a relation given as ordered pairs, the domain is the set of first coordinates and the range is the set of second coordinates. Each value is listed once.
A function is the rule that pairs each input with one output; the domain and range are the sets of inputs and outputs it uses.
Quick check: 3 questions
Question 1 of 3
What is the range of {(−3, 0), (1, 2), (4, 2)}?
Show answer and explanation
Answer: B. The range is the set of second coordinates, with repeats listed once: {0, 2}. Choice A is the domain, choice C repeats the 2, and choice D mixes inputs and outputs.
Question 2 of 3
What is the domain of f(x) = √x − 5?
Show answer and explanation
Answer: D. The expression under the root must be at least 0: x − 5 ≥ 0, so x ≥ 5. Choice B flips the sign of 5, and choices A and C allow inputs like 2 that give the square root of a negative.
Question 3 of 3
What is the range of y = x2?
Show answer and explanation
Answer: A. Squares are never negative, and 02 = 0, so the outputs are y ≥ 0. All real numbers is the domain, choice C describes x instead of y, and choice D leaves out the output 0.
Domain and Range: frequently asked questions
How do you find the domain and range from a graph?
For the domain, scan from left to right and note the smallest and largest x-values the graph covers; for the range, scan from bottom to top for the y-values. Closed dots include an endpoint, open dots leave it out, and arrows mean the graph keeps going. A line with arrows on both ends has domain and range all real numbers, unless it is horizontal.
Which is x and which is y, domain or range?
Domain is x and range is y. A memory trick: domain and range are in alphabetical order, the same as x and y, and inputs come before outputs. So for the point (6, 2), the 6 belongs to the domain and the 2 belongs to the range.
Can the domain be all real numbers?
Yes. Linear functions like y = 3x − 1, quadratics like y = x2 and every other polynomial accept any real input, so their domain is all real numbers. Restrictions appear only with denominators (which cannot be 0), even roots (which need a nonnegative inside), logarithms, or limits set by the real situation.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-IF.A.1, CCSS HSF-IF.B.5
Cite this entry
Effortless Math Team. “Domain and Range.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/domain-and-range/
Flashcard
TermDomain and Rangeinputs and outputs
DefinitionDomain: all inputs (x) a function accepts. Range: all outputs (y) it produces. For (1, 4), (2, 5): domain {1, 2}, range {4, 5}.
FunctionA 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 2DomainThe 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 2RelationA relation is any set of ordered pairs that links inputs to outputs, such as {(1, 4), (2, 5), (2, 7)}. It can be shown as a list, table, mapping diagram or graph. A relation in which every input has exactly one output is a function; this one is not, because 2 has two outputs.Functions · Grade 8, Algebra 1Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7InequalityAn 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, PrecalculusSquareA 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 3Real 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 1DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5Square 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 1DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1NumberA 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 7Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7Function NotationFunction notation is the way of writing a function as f(x), read “f of x”: f names the function, x is the input, and f(x) is the output. Writing f(x) = 2x + 5 means the same as y = 2x + 5, and f(3) = 11 says that the input 3 gives the output 11.Functions · Grade 8, Algebra 1, Algebra 2Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1InputAn input is the value you put into a function or a rule. The rule acts on the input to produce the output: for f(x) = x + 2, the input 5 gives the output 7. For many functions, the input is the x-value. In an input-output table, the inputs fill the first column.Functions · Grade 6, Grade 7, Grade 8OutputAn 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 8Independent VariableAn independent variable is an input value that can be chosen or changed. The dependent variable changes based on it. In y = 3x, x is the independent variable and y is the dependent one. On a graph the independent variable goes on the horizontal axis; in an experiment it is the factor you change on purpose.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 8