Function Formula

A function is a rule that assigns to each input in the domain exactly one output in the codomain — every input maps to precisely one output, never two.

The Formula

y=f(x)y = f(x)

When to use: A machine: put something in, get exactly one thing out. Same input always gives same output.

Quick Example

f(x)=x2f(x) = x^2 Input 3, output 9. Input −3-3, output 9. Never two different outputs for one input.

Notation

f(x)f(x) denotes the output of function ff at input xx. Also written f ⁣:X→Yf\colon X \to Y.

What This Formula Means

A function is a rule that assigns to each input in the domain exactly one output in the codomain — every input maps to precisely one output, never two.

A machine: put something in, get exactly one thing out. Same input always gives same output.

Formal View

f ⁣:X→Yf\colon X \to Y is a function   ⟺  \iff ∀x∈X,  ∃! y∈Y:(x,y)∈f\forall x \in X,\; \exists!\, y \in Y: (x, y) \in f

Worked Examples

Example 1

easy
Determine whether the relation {(1,3),(2,5),(3,7),(4,9)}\{(1, 3), (2, 5), (3, 7), (4, 9)\} is a function.

Answer

Yes, it is a function.\text{Yes, it is a function.}

First step

1
A relation is a function if every input (first element) maps to exactly one output (second element).

Full solution

  1. 2
    List the inputs: 1,2,3,41, 2, 3, 4. Each input appears exactly once, so each has a unique output.
  2. 3
    Since no input is repeated with a different output, this relation is a function.
A function assigns exactly one output to each input. The vertical line test for graphs and the uniqueness of first elements in ordered pairs are equivalent ways to check this.

Example 2

medium
Does the equation x2+y2=25x^2 + y^2 = 25 define yy as a function of xx?

Example 3

medium
Given f(x)=2x2−3x+1f(x) = 2x^2 - 3x + 1, find f(4)f(4) and determine if g={(1,2),(3,4),(1,5)}g = \{(1,2), (3,4), (1,5)\} is a function.

Common Mistakes

  • Calling any equation in xx and yy a function - check first that each xx gives only one yy (vertical line test).
  • Thinking one output can come from only one input - functions allow many inputs to share an output; they forbid one input giving many outputs.
  • Reading f(x)f(x) as ff times xx - it means the output of rule ff at input xx.

Why This Formula Matters

Function is the foundational object of all of advanced math: domain, range, inverses, composition, and calculus all assume the one-input-one-output rule. A student who lets one input produce two outputs builds every later concept on a broken foundation. Recognizing it by "Does every allowed input give exactly one output, never two?" — rather than by familiar numbers — is what lets a student tell it apart from relation and equation and variable in a mixed problem set.

Frequently Asked Questions

What is the Function formula?

A function is a rule that assigns to each input in the domain exactly one output in the codomain — every input maps to precisely one output, never two.

How do you use the Function formula?

A machine: put something in, get exactly one thing out. Same input always gives same output.

What do the symbols mean in the Function formula?

f(x)f(x) denotes the output of function ff at input xx. Also written f ⁣:X→Yf\colon X \to Y.

Why is the Function formula important in Math?

Function is the foundational object of all of advanced math: domain, range, inverses, composition, and calculus all assume the one-input-one-output rule. A student who lets one input produce two outputs builds every later concept on a broken foundation. Recognizing it by "Does every allowed input give exactly one output, never two?" — rather than by familiar numbers — is what lets a student tell it apart from relation and equation and variable in a mixed problem set.

What do students get wrong about Function?

The procedure for function is the easy part; the trap is calling any equation in xx and yy a function. Asking "Does every allowed input give exactly one output, never two?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Functions and Graphs: Complete Foundations for Algebra and Calculus →
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