The Function Concept• DEFINITION: A function consists of two

nonempty sets X and Y and a rule f that associates each element x in X with one and only one element y in Y.

Read “The function f from X into Y” andsymbolized by

f : X Y.

X Yf

The function f from X into Y

F “maps” X into Y

Some examples:•Supermarket item price

•Student chair

•College student GPA

•Worker SSN

•Car license plate “number”

•Real number x x2

More examples: Are these functions???

X Y• Dormitory rooms StudentsRule: room student(s) assigned

• Airplane luggage PassengersRule: piece(s) of luggage passenger

• Nine digit numbers WorkersRule: number worker’s SSN

• Real numbers Real numbersRule: x the numbers y such that y2= x

Another defintion:

Let X and Y be sets. A function f from X into Y is a set S of ordered pairs (x,y), x X, y Y, with the property that (x1, y1) and (x1, y2) are in S if and only if y1 = y2.

Examples

)}6,3(),4,3(),2,1(),1,2(),1,1(),0,0{()2(

)}10,3(),10,3(),5,2(),5,2(),2,1(),2,1(),1,0{()1(

S

S

Some Terminology & NotationLet f : X Y.

The set X (the “first” set) is called the domainof the function.

The set of y’s in Y which correspond to an element x in X is called the range of the function. The range of f is, in general a subset of Y.

Variables:

Let f : X Y.

The symbols x and y are called variables.

In particular, a symbol such as x, representing an arbitrary element in the domain is called anindependent variable.

A symbol such as y, representing an element in the range corresponding to an element x in the domain is called a dependent variable.

Function notation:

Let f : X Y.

Pick an element x in X and apply the rule f.This produces a unique element in Y. The symbol f(x) is used to denote that element.

f(x) is read “f of x” or “the value of f at x”or “the image of x under f .

Another picture

X Y

x f(x)f

More pictures

X Yf

f(X)

“Black box”x

f(x)f

One-to-one functions:Let f : X Y.

f is a one-to-one function if it takes distinctelements in the domain to distinct elements in the range. That is: f is one-to-one if

x1 x2 implies f(x1) f(x2).

Notation: f is 1 – 1.

Examples: Which of these function is 1 – 1?

• Supermarket item price

• Student GPA

• Car license plate “number”

• f(x) = 2x + 3

Inverse functionsSuppose f : XY is 1 – 1. Then there is a function g: f(X)X such that g(f(x)) = x for all x X.

g is called the inverse of f and is denoted by f -1

X Y

f(X)

f

g

Functions in Mathematics• From Geometry and Measurement:

1. Length function: x is a line segment, l(x) = the length of x.

2. Area functions: x is a rectangle,A(x) = the area of x.

3. Volume functions: x is a sphere,V(x) = the volume of x.

• From Probability & Statistics:E is a subset (event) in a sample space S,P(E) = the probability that E “occurs”.

Functions in “Algebra”

Let f : X Y where X is a given set of real numbers and Y is the set of all real numbers.

“f is a real-valued function of a real variable”

Note: The domain X may or may not be the set of all real numbers.

Examples:

Graph of a functionLet f : X Y. The graph of f is the set of points (x, f(x)) plotted in the coordinate plane:

Graph of f = {(x, f(x)) | x X }.

The graph of f is a “geometric” object – a “picture” of the function.

Examples:

1)( 2 nnf

1)( 2 xxf

1)( 2 xxf

1)( 2 xxf

Functions defined on the positive integers: Sequences

A function f whose domain is the set of positive integers is called a sequence.

The values are called the terms of the sequence; f(1) is the 1st term, f(2) is the 2nd term, and so on

(1) , (2) , (3) , , ( ) ,f f f f n

Subscript notationIt is customary to use subscript notation rather than functional notation:

and to denote the sequence by an

1 2 3(1) , (2) , (3) , , ( ) ,nf a f a f a f n a

Examples

1 3 4 5; first four terms: 2, , , ,2 3 4n

nan

2

1 1 2 3 4; first four terms: , , ,2 5 10 17n

nan

( 1) ; first four terms: 1, 1, 1, 1nna

2( 1) ; first four terms: 1, 4, 9, 16nna n

Recursion formulasA recursion formula or recurrence relation gives ak+1 in terms of one or more of the terms am that precede ak+1.

Examples: Find the first four terms and the nth term for the sequence specified by

1 1(1) 3 and 2 , 1,2,3,k ka a a k

1 1(2) 1 and ( 1) , 1, 2,3,k ka a k a k

Solutions

2 1

23 2

2 34 3

1

(1) 2 2 3,

2 2 2 3 2 3,

2 2 2 3 2 3

In general, 2 3nn

a a

a a

a a

a

2 1

3 2

4 3

(2) 2 2 1,3 3 2 1,4 4 3 2 1

In general, !n

a aa aa a

a n

More examples

(3) List the first six terms of the sequence whose nth term an is the nth prime number. Give a “formula” for an.

(4) The first four terms of the

sequence an are:

What is the 5th term?

1 1 11, , ,2 3 4

Answers(1) 2, 3, 5, 7, 11, 13; an = ??????

(2) 2

1201

48)4)(3)(2)(1(1 nnnn

nan

Limits of sequencesGiven a sequence an. What is the behavior of an for very large n ? That is, as n what can you say about an ?

Examples:

2

2

1 1(1) (2)

(3) ( 1) (4) ( 1)

n n

n nn n

n na an n

a a n

Answers

(1) 1 (2) 0

(3) No limit (4) No limit

Two special sequences

1. Arithmetic sequences: A sequence is an arithmetic sequence (arithmetic progression) if successive terms differ by a constant d, called the common difference. That is an is an arithmetic sequence if

1 for every positive integer k ka a d k

ExamplesDetermine whether the sequence is an arithmetic sequence(1) 2, 5, 8, 11, ,3 1(2) 1, 4, 9, 16,(3) 22, 18, 14, 10, ,

n

Answers:(1) Yes(2) No(3) Yes, assuming the pattern goes on as indicated

(4)What is the 12th term of the arithmetic sequence whose first three terms are:

1, 5, 9?

1 2 3(5) The sequence , , , is an arithmetic sequence.What is for all ?n

a a aa n

Solving the recursion formula

454)11(1 is (4) ofsolution The (5). solves This)1(

32

Thus,. implieswhich know We

12

1

134

123

12

11

adnaa

dadaadadaa

daa

daadaa

n

kkkk

Geometric sequences

A geometric sequence is a sequence in which the ratio of successive terms is a nonzero constant r. That is,

The number r is called the common ratio.

kkk

k arara

a

1

1 implieswhich

Examples(1)The sequence 8, 4, 2, 1, …. is a geometric

sequence. Find the common ratio and give the 5th term.

(2) The sequence

is a geometric sequence, find the common ratio and give the 6th term.

(3) an geometric sequence with common ratio r. Give a formula for an.

,85,

45,

25,5

Answers:

111

6

5

Then.Let )3(

325;

21)2(

21;

21)1(

nn raaaa

ar

ar

Function defined on intervalsLet f : X Y where X is an interval or a union of intervals and Y is the set of real numbers.

The graph of f is the set of all points (x,f(x)) in the coordinate plane.

The graph of f is the graph of the equation y=f (x).

Examples

f(x) = 2x + 1 f (x) = x2 + 1

The Elementary Functions

1. The constant functions: constant. a is where)( ccxf

The graph of f is a horizontal line c units above or below the x-axis depending on the sign of c.

f (x) = 2

(2) The identity function and linear functions

(a) The function f (x) = x is called the identity function. The graph is

.intercept- and slope with linestraight a is

ofgraph The . called are0,)(

form theof Functions (b)

bym

ffunctionslinearmbmxxf

12)( xxf

NONLINEAR FUNCTIONS

parabola. a isfunction quadratic aofgraph The . called are0 with constants are and , where

)(

form theof Functions (3)2

functionsquadraticacba

cbxaxxf

a > 0 a < 0

like looks cubic a ofgraph The . called are

0 with constants are and,,, where)(

form theof Functions (4)23

functionscubicadcba

dcxbxaxf

a > 0 a < 0

(5) Polynomial Functions

ts.coefficien theare,, ;polynomial theof deg theis

function. polynomial a is,0 with constants are,,,

andinteger enonnegativ a is where,)(

form theoffunction A

10

10

011

1

aareen

aaaan

axaxaxaxf

nn

nn

nn

converselynot but 1

)()( :function rational a isfunction polynomialA

on so andfunctions. quadratic ;2 degree of spolynomial :2

functions.linear ;1 degree of spolynomial :1

functions. constant nonzero the;0 degree of spolynomial :0

xpxp

n

n

n

(6) Rational functions

.1

)()(function rational a isfunction polynomialA

functions. polynomial are and where)()()(

form theoffunction a is A

xpxp

qpxqxpxr

unctionrational f

Some graphs

21)(

x

xf1

)( 2

xxxf

112)( 2

xxxf

The Elementary Functions

(7) Algebraic functions: sums, differences, products, quotients and roots of rational functions.

(8) The trigonometric functions.(9) Exponential functions.(10) Logarithm functions.