Introduction to Calculus: Functions

Functions

A function f is a rule that assigns an element f(x) of a set, called the target set of the function f, to any element x of the domain of definition of the function. The symbol x is called the variable of the

function f.

DefinitionDefinition

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

Domain of Definition of a Function

The set of values of the variable(s) of a function f for which the function is

defined is called the domain of definition of the function

f.

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

A Thermometer is a Function

This picture shows a reconstruction of a thermometer of Galileo. The

sealed glass cylinder contains clear liquid and objects which sink

differently as the cylinder is heated.

The variable of the thermometer function is time, and the value of

the function is the current temperature.

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

Computer Programs are Functionsfunction putDataXML(xmlDoc,sDocPath)

{if(gnLoadFts==1){

var node=xmlDoc.lastChild;if(node){

var oChild=node.firstChild;

var aFCD=new Array();var aFTCD=new Array();while(oChild){

if(oChild.nodeName=="chunkinfo"){

.....

A computer program produces

an output of a given input, and is,

therefore, a function. Even more, programs use subroutines

that are functions themselves.

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

Functions defined by Tables

A function whose domain of definition is a

finite set can also be defined by a table from which we can read the values of the function

for all possible values of the variable.

x f(x)

a α

b β

c γ

d δ

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

Functions in Calculus

Functions in calculus are usually defined by algebraic expressions.

The function f, ,

is defined for all x ≠ 1.

f x( ) =

x2

x −1

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

Graphs of FunctionsA function f, whose domain of definition is a subset of the set of real numbers and

whose values are real numbers, can

be pictured by drawing the points (x, f(x)) for some

values of x. x

f(x)

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

Notations of FunctionsThe notation

f: A →Bindicates that the domain of the

definition of the function f is the set A, and that to each element of the set A the function f assigns an element of

the set B. The set B is the target set of the function f.

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

Functions

It defines, therefore, a function, whose variable takes values in the interval [-1,1].

The values of this function are real numbers.

The expression takes real values if -1 ≤ x ≤ 1.

1−x2

f x( ) = 1−x2

We indicate this by writing

f: [-1,1]→R, .

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

FunctionsThe function f: A → B assigns, to any

element a of the set A an element f(a) of the set B.

The set A is the domain of definition of the function f, the set B is the target set of the function

f.

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

The variable a of the function f varies in the set A. For any a ∈ A, f(a) ∈ B.

History of Functions

The concept of a function was defined by Leonhard Euler in the

mid 18th century.

Using functions one can develop general theories.

Preliminaries/Introduction to Calculus/Functions by M. Seppälä

Leonhard Euler (1707 - 1783)