An 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.
The input goes in, the rule acts on it, and one output comes out. In a table the input is the first column; in an ordered pair or f(5), the input is the number written first or inside the parentheses.
Formal definition and notation
For a functionf, an input is an element x of its domain; the function assigns to it exactly one output, written f(x). In the ordered pair(x, f(x)) the input is the first coordinate, and on a graph it is read on the horizontal axis. The set of all allowed inputs is the domain; a value such as x = 2 for f(x) = 1x − 2 is not an allowed input, because the rule cannot produce an output for it.
Symbol
Read it as
Meaning
f(5)
f of 5
the output when the input is 5
(5, 7)
the point five, seven
an input-output pair; the input is listed first
x → 2x
x maps to two x
the input x is sent to the output 2x
Key facts
Each input has exactly one output in a function. Two different inputs may share an output.
In f(x), the value inside the parentheses is the input: f(4) means “use 4 as the input”.
Finding the output from an input is evaluating. Finding the input from an output is solving: if 2x = 10, the input is 5.
Some values cannot be inputs. You cannot divide by zero, so 3 is not an allowed input for 1x − 3.
Try it: guess the rule
Interactive
The rule is hidden. Type an input and press Run (or Enter), then try more inputs and study the table of input-output pairs. When you think you know the rule, press Show rule to check.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Input examples, step by step
Example 1: Output from an input
Warm-up
For f(x) = x2 + 1, what is the output when the input is −3?
Put the input in place of x: f(−3) = (−3)2 + 1.
(−3)2 = 9, because a negative times a negative is positive.
9 + 1 = 10.
Answer: The output is 10, so f(−3) = 10.
Example 2: Find the rule from a table
Core
An input-output table has inputs 1, 2, 3, 4 and outputs 7, 11, 15, 19. Find the rule, then the output for the input 10.
Each time the input goes up by 1, the output goes up by 4, so the rule starts with 4x.
For input 1, 4 × 1 = 4, but the output is 7, so add 3: the rule is 4x + 3.
Check another row: 4 × 3 + 3 = 15. It matches.
Input 10: 4 × 10 + 3 = 43.
Answer: The rule is 4x + 3, and the input 10 gives the output 43.
Example 3: Find the input from an output
Test-style
For g(x) = 5x − 8, which input gives an output of 27?
This time the output is known: set 5x − 8 = 27.
Add 8: 5x = 35.
Divide by 5: x = 7.
Check: g(7) = 35 − 8 = 27. Trap: g(27) = 127 answers a different question.
Answer: The input is 7.
Common mistakes
✗ MistakeA table with the input 3 paired with 8 and also with 10 describes a function.
✓ InsteadIt does not: the input 3 has two different outputs.
A function can have only one output for each input. Repeated inputs must always lead to the same output.
✗ MistakeInput 2 and output 9 make the ordered pair (9, 2).
✓ InsteadThe pair is (2, 9): input first, output second.
Ordered pairs list the input (x-value) first. Swapping them plots a different point.
✗ Mistake“Find f(4)” for f(x) = 2x: solve 2x = 4, so the answer is 2.
✓ Insteadf(4) means the input is 4: f(4) = 2 × 4 = 8.
f(4) asks for an output. f(x) = 4 would ask for an input. Look at where the 4 sits.
✗ MistakeFor h(x) = 6x − 2, the input 2 gives the output 0.
✓ Instead2 is not an allowed input: it makes the denominator2 − 2 = 0.
Division by zero is undefined, so 2 is left out of the domain. Check that an input makes sense before you use it.
The domain is the set of all allowed inputs. One input is a single member of the domain.
Quick check: 3 questions
Question 1 of 3
For f(x) = 3x − 2, what is the output when the input is 6?
Show answer and explanation
Answer: C.f(6) = 3(6) − 2 = 18 − 2 = 16. 20 adds 2 instead of subtracting, 83 solves 3x − 2 = 6 (treating 6 as the output), and 6 just repeats the input.
Question 2 of 3
For h(x) = 2x + 5, which input gives an output of 11?
Show answer and explanation
Answer: B. Solve 2x + 5 = 11: 2x = 6, so x = 3. 27 is h(11), the output for input 11; 6 forgets to divide by 2; and 11 is the output itself.
Question 3 of 3
The point (−2, 7) is on the graph of a function. What is the input?
Show answer and explanation
Answer: A. In an ordered pair the input comes first, so the input is −2 and the output is 7. Choices C and D add and multiply the coordinates, which has no meaning here.
Input: frequently asked questions
What is the difference between an input and an output?
The input is the number you start with; the output is the number the rule gives back. For the rule “add 5”, the input 3 produces the output 8. In y = x + 5, x is the input and y is the output, and in the ordered pair (3, 8) the input comes first.
Is the input always x?
No. x is the most common name, but any letter works. In d = 60t the input is time t; in C(n) = 4n it is the number of items n. Find the input by asking which quantity you choose first, or which letter sits inside the parentheses in function notation.
What is an input-output table?
It is a table with the inputs in one column (or row) and the matching outputs beside them. It lets you see a rule at work and find patterns. If the inputs 1, 2, 3 give 5, 8, 11, the outputs grow by 3 each time, so the rule is 3x + 2. Each input should appear with only one output.
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 2OutputAn 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 8Ordered 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 7Independent 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 8DenominatorThe 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 5DivisionDivision 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 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 2NumberA 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 2Function 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 2VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7