What Is a Function?

Introduction

Functions are the backbone of all of mathematics and science. A function is a rule that assigns exactly one output to each input. Understanding functions unlocks calculus, statistics, physics, and computer science.

The Definition of a Function

A function is a relation in which each element of the domain (input) corresponds to exactly ONE element of the range (output). Think of it like a machine: put something in, get exactly one thing out. No input can produce two different outputs.

A simple test: if you can find ANY input value that produces TWO different outputs, it is NOT a function.

Domain and Range

The domain is the set of all valid inputs. The range is the set of all resulting outputs. When finding the domain, ask: "What values of x are NOT allowed?" Common restrictions include: division by zero (denominator cannot be 0) and even roots of negative numbers.

f: Domain → Range

Key Terms Glossary

Function

A relation where each input (x-value) maps to exactly one output (y-value).

Domain

The set of all valid input values for a function.

Range

The set of all output values produced by a function.

Vertical Line Test

A graphical test: if any vertical line crosses the graph more than once, it is not a function.

Quick practice

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0 / 1 answered (demo)

Which of the following is a function? Consider the relation as a set of (x, y) pairs.
Select one option{(1,2), (1,3), (2,4)}{(1,2), (2,3), (3,4)}{(2,1), (2,3), (4,2)}{(0,0), (0,1), (1,0)}

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