The 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.
The domain is every input that gives a real output. A square root starts at a boundary; a fraction loses the value that makes its denominator 0.
Formal definition and notation
For a functionf from a set A to a set B, the domain is the set A of inputs; each element of the domain is assigned exactly one output. When a function is given only by a formula, its implied domain is the set of all real numbersx for which f(x) is a real number. In a modeling problem, the domain is further limited to inputs that make sense in the context.
A theater has 120 seats. Ticket revenue is R(n) = 15n dollars when n tickets are sold. What is a reasonable domain, and how many values does it contain?
The formula works for any real n, but the situation does not.
Tickets come in whole numbers, and n cannot be negative or more than the 120 seats.
Domain: the whole numbers 0, 1, 2, …, 120.
Count: from 0 to 120 there are 120 − 0 + 1 = 121 values.
Answer: The domain is the whole numbers from 0 to 120, which is 121 values.
Common mistakes
✗ Mistakeh(x) = 7x − 5 has domain x ≠ 7.
✓ InsteadSet the denominator to 0: x − 5 = 0, so the domain is x ≠ 5.
Only the denominator causes the restriction. The numerator can be any number, including 0.
✗ MistakeFor √x + 4, the domain is x > 0.
✓ Insteadx + 4 ≥ 0, so x ≥ −4.
It is the whole radicand that must be nonnegative, not x itself. At x = −4, √0 = 0 is allowed.
✗ MistakeFor the points (2, 5), (3, 8), (7, 5), the domain is {5, 8}.
✓ InsteadThe domain is the set of first coordinates: {2, 3, 7}.
Domain goes with inputs, the x-values. The y-values form the range.
The range of a function is its set of outputs. For y = x2 the domain is all real numbers, but the range is only y ≥ 0. (In statistics, range means max minus min, a different idea.)
That entry treats inputs and outputs together. This one focuses on the input side and the rules that remove inputs.
Quick check: 3 questions
Question 1 of 3
What is the domain of f(x) = 2x + 6?
Show answer and explanation
Answer: C.x + 6 = 0 when x = −6, so that input is removed. x ≠ 6 has the wrong sign, x ≠ 2 looked at the numerator, and all real numbers ignores the division by zero.
Question 2 of 3
What is the domain of g(x) = √x − 8?
Show answer and explanation
Answer: A. Solve x − 8 ≥ 0: x ≥ 8. 8 is included because √0 = 0. x ≥ 0 checked x instead of the radicand, x > 8 wrongly excluded 8, and x ≤ 8 flipped the inequality.
Question 3 of 3
Which function has a domain of all real numbers?
Show answer and explanation
Answer: D. A polynomial can be evaluated at any real number, so y = x2 − 4x has no restrictions. 1x excludes 0, and the two square roots exclude inputs that make the radicand negative.
Domain: frequently asked questions
How do you find the domain of a function?
Start with all real numbers and remove what breaks a rule. Set any denominator equal to 0 and exclude those values. Require any even root’s radicand to be 0 or more and solve that inequality. If the function models a real situation, also keep only inputs that make sense, such as nonnegative times or whole numbers of items.
How do you find the domain from a graph?
Look left to right along the x-axis and ask which x-values have a point of the graph above or below them. The leftmost and rightmost points give the endpoints: a closed dot includes the value, an open dot leaves it out, and an arrow means the graph continues forever in that direction. Gaps or holes are left out too.
What is the domain of a linear function?
All real numbers, written (−∞, ∞). A line such as y = 2x − 3 can take any input, because multiplying and adding never fail. The only exception is a word problem, where the context may limit the inputs, for example to times from 0 to 10 hours.
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 2Real 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, PrecalculusOrdered 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 7Range of a FunctionThe 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.Functions · Grade 8, Algebra 1, Algebra 2DenominatorThe 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 5Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5NumberA 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 2NonnegativeNonnegative means greater than or equal to zero: such a number is either 0 or positive, so x ≥ 0. The nonnegative integers are 0, 1, 2, 3 and so on. It is not the same as positive, because 0 is nonnegative but not positive. Distances, absolute values and principal square roots are always nonnegative.Problem Solving · Grade 6, Grade 7, Algebra 1Domain 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 2SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7InputAn 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 8RelationA 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 1Piecewise FunctionA piecewise function is a function defined by two or more formulas, each used on its own interval of the domain. To find an output, first check which interval the input belongs to, then use that piece’s formula. Absolute value, step functions and tiered prices are common examples.Functions · Algebra 1, Algebra 2, PrecalculusIndependent 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 8