An 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.
In Spanish
salida
Also called
y-value, dependent value, f(x)
Grade level
6-8
Grade 6Grade 7Grade 8Algebra 1CCSS 8.F.A.1SATACT
Output in a picture
The rule turns each input into exactly one output: 4 × 2 + 1 = 9. In a table, the outputs are the second column.
Formal definition and notation
For a functionf and an input x in its domain, the output is f(x), the single value that the function assigns to x. A function gives exactly one output for each input, although different inputs may share the same output. The set of all outputs is the range of the function. When a function is written y = …, the output is the value of the dependent variabley, and in an ordered pair(x, y) it is the second coordinate.
Symbol
Read it as
Meaning
f(x)
f of x
the output of the function f for the input x
f(3) = 7
f of 3 equals 7
the input 3 gives the output 7
y
y
the usual letter for the output when a rule is written y = …
Key facts
A function gives each input exactly one output; that is what makes it a function.
Different inputs can share an output: for y = x2, the inputs 3 and −3 both give 9.
The output is the dependent variable: its value depends on the input you choose.
On a graph, inputs are read on the horizontal axis and outputs on the vertical axis.
Working backward from an output to the input means solving an equation: if 2x + 1 = 15, the input was 7.
Finding an output
Finding an output
output = f(input)
input
the value you choose or are given, usually x
f
the rule: an equation, a table, a graph or words
Substitute the input for every x in the rule, then simplify using the order of operations. For f(x) = 4x − 3, f(5) = 4(5) − 3 = 17.
Try it: run the machine
Interactive
Type an input and press Run to see its output; the table keeps your last few results. Press Hide rule, then try to work out the rule from the outputs alone.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Output examples, step by step
Example 1: Substitute the input
Warm-up
For f(x) = 4x + 1, what is the output when the input is 5?
Replace x with 5: f(5) = 4(5) + 1.
Multiply first: 4 × 5 = 20.
Add: 20 + 1 = 21.
Answer: The output is 21, so f(5) = 21.
Example 2: Find the rule from a table
Core
A table has inputs 1, 2, 3, 4 and outputs 7, 10, 13, 16. What is the output for the input 10?
Each time the input goes up by 1, the output goes up by 3, so the rule starts with 3x.
For input 1, 3 × 1 = 3, but the output is 7, so add 4: the rule is y = 3x + 4.
Check another row: 3 × 4 + 4 = 16. It works.
Input 10: 3 × 10 + 4 = 34.
Answer: The rule is y = 3x + 4, so the output is 34.
Example 3: Outputs in function notation
Test-style
Let f(x) = x2 − 3x. What is f(−2), and which inputs give an output of 0?
The input is the value that goes in (usually x); the output is what comes out (usually y or f(x)). For y = x + 5, input 2 gives output 7.
Outputvs.Range of a function
The range is the set of all possible outputs, not one value. For y = x2, each output is a single number like 9, and the range is all numbers 0 or greater.
The dependent variable is the letter or quantity that holds the outputs, like the total cost. An output is one value of it, like 24 dollars.
Quick check: 3 questions
Question 1 of 3
For y = 5x − 2, what is the output when the input is 3?
Show answer and explanation
Answer: B.5 × 3 − 2 = 15 − 2 = 13. Choice A forgets to subtract 2, C computes 5(3 − 2), and D runs the rule backward: (3 + 2) ÷ 5.
Question 2 of 3
A rule gives these input-output pairs: 2 → 8, 5 → 20, 7 → 28. What is the output for the input 10?
Show answer and explanation
Answer: C. Each output is 4 times its input, so the rule is y = 4x and the output is 40. Choice A adds the last jump of 8 to 28 again, B adds 4 instead of multiplying by 4, and D multiplies by 3.
Question 3 of 3
Which statement is true for every function?
Show answer and explanation
Answer: D. A function assigns exactly one output to each input. Outputs can repeat: for y = x2, 3 and −3 both give 9, so choices A and C are false. Outputs can also be smaller than inputs, as in y = x − 5.
Output: frequently asked questions
Is the output x or y?
Usually y. When a rule is written y = 2x + 1, you choose x (the input) and compute y (the output). In function notation the output is written f(x), read f of x, so f(4) = 9 means the input 4 has the output 9. On a graph, outputs are measured on the vertical y-axis.
Can two inputs have the same output?
Yes. A function only forbids one input from having two different outputs. Many functions repeat outputs: y = x2 gives 16 for both 4 and −4, and a flat line such as y = 5 gives the output 5 for every input. On a graph this shows up as a horizontal line crossing the curve more than once.
What is an input-output table?
It is a two-column table that lists inputs in the first column and the matching outputs in the second. Each row is an ordered pair. To find the rule, compare how the outputs change as the inputs change; if the outputs go up by the same amount each step, the rule is linear, like y = 3x + 4.
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 2Dependent 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 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 7SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8NumberA 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 2InputAn 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 8Function 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 2Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7VariableA 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 8ExpressionAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Algebra · 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 2GraphA 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 7Linear 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 1Independent 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