Functions

Introduction

A relation is a set of input values and output values, often written as an ordered pair of the form (input, output). A function is a relation in which each possible input value leads to exactly one output value. The input values make up the Domain of a Function, and the output values make up the Range of a Function, also known as the codomain. A relation is NOT a function if one input in the domain maps to multiple outputs in the range. For example, consider the relations R1 = {(1,2), (1, 3), (2, 4), (3, 5)} and R2 = {(1,2), (2, 4), (3, 5), (4, 5)}. The relation R1 is not a function because the input 1 maps to two different outputs: 2 and 3. However, the relation R2 is a function since each input only maps to one output. Note that multiple inputs can map to a single output; that is, R2 is still a function despite the fact that the inputs 3 and 4 both map to the same output, 5. If a function maps each input to a distinct output (that is, no two inputs lead to the same output), we say that the function is "one-to-one" or "injective". For example, R3 = {(-1, 0), (2, 5), (3, 10)} is a one-to-one function since each input maps to a unique output.

Resources

• What is a Function?, Khan Academy
• Relations vs. Functions, Cliff's Notes
• Determining if a Relation is a Function, Lumen Learning
• Identifying Functions Using Graphs, Lumen Learning