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)

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

Quick Example

f(x)=x2 Input 3, output 9. Input −3, output 9. Never two different outputs for one input.

Notation

f(x) denotes the output of function f at input x. Also written f ⁣:X→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→Y is a function   ⟺   ∀x∈X,  ∃! y∈Y:(x,y)∈f

Worked Examples

Example 1

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

Answer

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: 1,2,3,4. 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=25 define y as a function of x?

Example 3

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

Common Mistakes

  • Calling any equation in x and y a function - check first that each x gives only one y (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) as f times x - it means the output of rule f at input x.

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) denotes the output of function f at input x. Also written f ⁣:X→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 x and y 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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