Difference between revisions of "Functions"

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==Introduction==
 
==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 following 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 both 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 mapping 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".
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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==
 
==Resources==

Revision as of 11:38, 15 September 2021

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