# What is Function in Math — Definition, Types & Examples

## TL;DR

A function in math is a rule that assigns to every input value exactly one output value — written y=f(x) where x is the input and y is the output. This article gives the formal definition, distinguishes a function from a general relation, covers domain and range, the vertical line test, the most-used function families, three worked examples, and the common confusions.

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### A function

is a relationship between two sets — call them inputs and outputs — such that every input maps to exactly one output. The output may be the same for different inputs, but no input is allowed to have two different outputs.

## The Formal Definition

A function f from a set A (the domain) to a set B (the codomain) is a rule that assigns to each element x ∈ A exactly one element f(x) ∈ B.

Three things must be true:

1. **Every input has an output.** No element of the domain is left unmapped.
2. **Each input has only one output.** No input maps to two different outputs.
3. **The domain and codomain are defined.** "What's allowed as input" and "what's allowed as output" are part of the function.

The notation is:
f:A→B, x ↦ f(x).

Read aloud: "f from A to B, x maps to f(x)."

### Examples:

- f(x) = 2x — input 3, output 6. Linear function.
- f(x) = x² — input -2, output 4. Quadratic function.
- f(x) = sin x — input 90°, output 1. Trigonometric function.

> **Quick reference.**  
> - **Definition:** a rule mapping each input to exactly one output.  
> - **Notation:** y=f(x) or f:A→B.  
> - **Domain:** the set of allowed inputs.  
> - **Range:** the set of actual outputs (a subset of the codomain).  
> - **Vertical line test:** a graph represents a function if and only if every vertical line meets it at most once.

## Function vs Relation — The Key Distinction

Every function is a relation; not every relation is a function.

A relation is any rule pairing elements of one set with elements of another. A relation becomes a function only when each input has exactly one output.

|  | Relation | Function |
| --- | --- | --- |
| Input rule | Any pairing | Each input has exactly one output |
| Visual test | None | Vertical line test |
| Examples | (1,2),(1,5),(2,7) | (1,2),(2,5),(3,7) |

A circle x²+y²=9 is a relation but not a function — for x=0, both y=3 and y=-3 are valid. A vertical line through x=0 crosses the circle twice.

## Domain and Range

The **domain** of a function f is the set of all valid inputs — the values of x for which f(x) is defined.

The **range** of f is the set of all actual outputs — the values f(x) takes as x varies across the domain.

For f(x)=√x: domain is x≥0 (you can't take real square roots of negatives); range is f(x)≥0.

For f(x)=1/x: domain is x≠0; range is f(x)≠0.

Finding the domain and range is the first step in graphing or analyzing any function.

## The Most-Used Function Families

| Family | Form | Example |
| --- | --- | --- |
| **Linear** | f(x)=mx+b | f(x)=3x+2 |
| **Quadratic** | f(x)=ax²+bx+c | f(x)=x²−5 |
| **Polynomial** | f(x)=aₙxⁿ+⋯+a₀ | f(x)=x³−4x+1 |
| **Rational** | f(x)=p(x)/q(x) | f(x)=1/(x−2) |
| **Exponential** | f(x)=a^x | f(x)=2^x |
| **Logarithmic** | f(x)=logₐx | f(x)=lnx |
| **Trigonometric** | f(x)=sin x, cos x, tan x | f(x)=sin x |
| **Absolute value** | f(x) = |x| |
| **Step / piecewise** | depends on the piece | ⌊x⌋ (floor) |

Every member of a family shares the same algebraic structure, distinguished by parameter values.

## Three Worked Examples of Function — Quick, Standard, Stretch

**Quick.** If f(x)=2x+5, find f(3).

Substitute x=3:
f(3)=2(3)+5=11. 
**Final answer:** f(3)=11.

**Standard (Wrong Path First — Where Functions Trip Students).** The relation (1,2),(2,5),(1,7),(3,9) — is this a function?

_A student counts four ordered pairs and says, "yes, it's a function."

The flaw: counting pairs doesn't check whether each input has only one output. Look more closely: the input 1 appears twice — once with output 2 and once with output 7.

_The rescue._ The same input 1 maps to two different outputs (2 and 7). Thus, it's not a function.

**Final answer:** Not a function (input 1 has two outputs).

**Stretch.** Find the domain of f(x)=1/(x−3).

Two restrictions to handle:
- The expression under the square root must be >0 (cannot be zero because it's in the denominator, and cannot be negative because of the real square root). So x−3>0, giving x>3.

**Final answer:** domain is x>3 (or (3,∞) in interval notation).

## Where Functions Appear — Beyond the Textbook

Functions describe every relationship in modeling.

- **Physics.** Position as a function of time, velocity as a function of position.
- **Economics.** Demand as a function of price.
- **Computer science.** Every line of code in `f(x) = x*2 + 3` is a function.
- **Biology.** Population size as a function of time.
- **Cryptography.** Modern internet security uses one-way functions.

The modern definition of a function — as a rule mapping inputs to outputs — was formalized in the 1800s by mathematicians including Peter Gustav Lejeune Dirichlet and Bernhard Riemann. Earlier figures had used the word "function" in a looser sense tied to specific formulas.

## Tripping Points to Avoid In Function

### **Mistake 1:** Treating any pairing as a function
Where it slips in: A list of ordered pairs with a repeated input is called a function.

**Don't do this:** Skip the "each input has one output" check.

### **Mistake 2:** Confusing range with codomain
Where it slips in: Student equates range with codomain.

**Don't do this:** Treat the codomain as the range.

### **Mistake 3:** Forgetting domain restrictions when computing
Where it slips in: Plugging in values without checking domain.

## Conclusion

- A function is a rule that assigns each input exactly one output.
- Notation y=f(x) with the input as x and output as f(x).
- The vertical line test is the visual check for function-ness.
- Domain === allowed inputs; range === actual outputs.

## Practice These Three Before Moving On

1. If f(x)=3x−4, find f(5).
2. Is the relation (1,4),(2,5),(3,4) a function?
3. Find the domain of f(x)=1/(x−7).

If problem 2 gave "no" because 4 appears twice as an output, return to FAQ "Can two different inputs give the same output?"