Function 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.
In f(3) = 11, f is the name, 3 is the input, and 11 is the output. Together they give a point on the graph.
Formal definition and notation
If f is a function and x is an element of its domain, then f(x) denotes the unique element of the range that f assigns to x. The equation y = f(x) states that y is the output of f at input x; the graph of f is the set of points (x, f(x)). The parentheses mark an input, not multiplication.
Symbol
Read it as
Meaning
f(x)
f of x
the output of function f when the input is x
f(3) = 11
f of 3 equals 11
input 3, output 11; the point (3, 11) is on the graph
y = 2x + 5 and f(x) = 2x + 5 describe the same line, but f notation also shows which input you used: f(3) = 11 instead of “when x = 3, y = 11“.
Quick check: 3 questions
Question 1 of 3
If p(x) = 6 − x2, what is p(−2)?
Show answer and explanation
Answer: B.p(−2) = 6 − (−2)2 = 6 − 4 = 2. 10 treated −x2 as +4, −12 multiplied p by −2 as if p = 6, and 8 added 2 to 6.
Question 2 of 3
The point (4, −1) is on the graph of f. Which statement is true?
Show answer and explanation
Answer: C. A point (x, y) on the graph means f(x) = y, so f(4) = −1. f(−1) = 4 swaps input and output, f(4) = −4 is not shown by the point, and f = −14 treats f(4) as multiplication.
Question 3 of 3
T(h) = 70 + 3h gives the temperature, in degrees, h hours after 6 a.m. What does T(2) = 76 mean?
Show answer and explanation
Answer: A. The input 2 is hours after 6 a.m., which is 8 a.m., and the output 76 is the temperature. The other choices treat the output as a change, swap the input and output, or forget that time starts at 6 a.m.
Function Notation: frequently asked questions
What does f(x) mean?
f(x) means the output of the function named f when the input is x. It is read “f of x”. If f(x) = x + 10, then f(x) is just another name for y, and f(5) = 15 is the output for the input 5. It never means f multiplied by x.
Why use f(x) instead of y?
Function notation shows the input and the output together. f(2) = 9 says in one short statement that the input 2 gives 9, which with y takes a sentence. It also lets you name several functions, such as f, g and h, and compare or combine them, as in f(2) + g(2).
Does the letter have to be f and x?
No. Any letters work. Scientists write h(t) for height after t seconds, and businesses write C(n) for the cost of n items. The letter outside the parentheses names the function, and the one inside stands for the input. Choosing letters that match the context makes the notation easier to read.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-IF.A.2
Cite this entry
Effortless Math Team. “Function Notation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/function-notation/
Flashcard
TermFunction Notationf(x)
DefinitionRead “f of x”: the output of function f at input x. If f(x) = 2x + 5, then f(3) = 11.
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 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4SubstituteSubstitute 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 8EvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1MileA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5, Grade 6Equal 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 2Square 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 1DataData 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 2EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6InputAn 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 8Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1Composition of FunctionsA composition of functions applies one function to the output of another. The composite f(g(x)), also written (f ∘ g)(x), means: find g(x) first, then put that result into f. If g(x) = x + 1 and f(x) = x2, then f(g(2)) = f(3) = 9. Order matters.Functions · Algebra 2, PrecalculusInverse FunctionAn inverse function undoes another function: if f sends a to b, its inverse f−1 sends b back to a. For f(x) = 2x + 3, the inverse is f−1(x) = x − 32. Only one-to-one functions have inverses, and the graph of f−1 is the mirror image of f across y = x.Functions · 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