functions

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… … and the following and the following mathematical appetizer is mathematical appetizer is

about…about…

FunctionsFunctions

Wishing you bon appétit:Wishing you bon appétit:

Elka VeselinovaElka Veselinova

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FunctionsFunctions

A A functionfunction f from a set A to a set B is an f from a set A to a set B is an assignmentassignment of exactly one element of B to each of exactly one element of B to each element of A.element of A.

We writeWe write

f(a) = bf(a) = b

if b is the unique element of B assigned by the if b is the unique element of B assigned by the function f to the element a of A.function f to the element a of A.

If f is a function from A to B, we writeIf f is a function from A to B, we write

f: Af: ABB

(note: Here, “(note: Here, ““ has nothing to do with if… “ has nothing to do with if… then)then)

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FunctionsFunctions

If f:AIf f:AB, we say that A is the B, we say that A is the domaindomain of f and B of f and B is the is the codomaincodomain of f. of f.

If f(a) = b, we say that b is the If f(a) = b, we say that b is the imageimage of a and of a and a is the a is the pre-imagepre-image of b. of b.

The The rangerange of f:A of f:AB is the set of all images of B is the set of all images of elements of A.elements of A.

We say that f:AWe say that f:AB B mapsmaps A to B. A to B.

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FunctionsFunctions

Let us take a look at the function f:PLet us take a look at the function f:PC withC withP = {Linda, Max, Kathy, Peter}P = {Linda, Max, Kathy, Peter}C = {Boston, New York, Hong Kong, Moscow}C = {Boston, New York, Hong Kong, Moscow}

f(Linda) = Moscowf(Linda) = Moscowf(Max) = Bostonf(Max) = Bostonf(Kathy) = Hong Kongf(Kathy) = Hong Kongf(Peter) = New Yorkf(Peter) = New York

Here, the range of f is C.Here, the range of f is C.

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FunctionsFunctions

Let us re-specify f as follows:Let us re-specify f as follows:

f(Linda) = Moscowf(Linda) = Moscow

f(Max) = Bostonf(Max) = Boston

f(Kathy) = Hong Kongf(Kathy) = Hong Kong

f(Peter) = Bostonf(Peter) = Boston

Is f still a function?Is f still a function? yesyes

{Moscow, Boston, Hong {Moscow, Boston, Hong Kong}Kong}

What is its range?What is its range?

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FunctionsFunctions

Other ways to represent f:Other ways to represent f:

BostonBostonPeterPeter

Hong Hong KongKongKathyKathy

BostonBostonMaxMax

MoscowMoscowLindaLinda

f(x)f(x)xx LindaLinda

MaxMax

KathyKathy

PeterPeter

BostonBoston

New New YorkYorkHong Hong KongKong

MoscowMoscow

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FunctionsFunctions

If the domain of our function f is large, it is If the domain of our function f is large, it is convenient to specify f with a convenient to specify f with a formulaformula, e.g.:, e.g.:

f:f:RRRR f(x) = 2xf(x) = 2x

This leads to:This leads to:f(1) = 2f(1) = 2f(3) = 6f(3) = 6f(-3) = -6f(-3) = -6……

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FunctionsFunctions

Let fLet f11 and f and f22 be functions from A to be functions from A to RR..

Then the Then the sumsum and the and the productproduct of f of f11 and f and f22 are are also functions from A to also functions from A to RR defined by: defined by:

(f(f11 + f + f22)(x) = f)(x) = f11(x) + f(x) + f22(x)(x)

(f(f11ff22)(x) = f)(x) = f11(x) f(x) f22(x)(x)

Example:Example:

ff11(x) = 3x, f(x) = 3x, f22(x) = x + 5(x) = x + 5

(f(f11 + f + f22)(x) = f)(x) = f11(x) + f(x) + f22(x) = 3x + x + 5 = 4x + 5(x) = 3x + x + 5 = 4x + 5

(f(f11ff22)(x) = f)(x) = f11(x) f(x) f22(x) = 3x (x + 5) = 3x(x) = 3x (x + 5) = 3x22 + 15x + 15x

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FunctionsFunctions

We already know that the We already know that the rangerange of a function of a function f:Af:AB is the set of all images of elements aB is the set of all images of elements aA.A.

If we only regard a If we only regard a subsetsubset S SA, the set of all A, the set of all images of elements simages of elements sS is called the S is called the imageimage of of S.S.

We denote the image of S by f(S):We denote the image of S by f(S):

f(S) = {f(s) | sf(S) = {f(s) | sS}S}

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FunctionsFunctions

Let us look at the following well-known function:Let us look at the following well-known function:





What is the image of S = {Linda, Max} ?What is the image of S = {Linda, Max} ?

f(S) = {Moscow, Boston}f(S) = {Moscow, Boston}

What is the image of S = {Max, Peter} ?What is the image of S = {Max, Peter} ?

f(S) = {Boston}f(S) = {Boston}

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Properties of FunctionsProperties of Functions

A function f:AA function f:AB is said to be B is said to be one-to-oneone-to-one (or (or injectiveinjective), if and only if), if and only if

x, yx, yA (f(x) = f(y) A (f(x) = f(y) x = y) x = y)

In other words:In other words: f is one-to-one if and only if it f is one-to-one if and only if it does not map two distinct elements of A onto does not map two distinct elements of A onto the same element of B.the same element of B.

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And again…And again…





Is f one-to-one?Is f one-to-one?

No, Max and Peter are No, Max and Peter are mapped onto the mapped onto the same element of the same element of the image.image.

g(Linda) = Moscowg(Linda) = Moscow

g(Max) = Bostong(Max) = Boston

g(Kathy) = Hong Kongg(Kathy) = Hong Kong

g(Peter) = New Yorkg(Peter) = New York

Is g one-to-one?Is g one-to-one?

Yes, each element is Yes, each element is assigned a unique assigned a unique element of the image.element of the image.

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How can we prove that a function f is one-to-How can we prove that a function f is one-to-one?one?

Whenever you want to prove something, first Whenever you want to prove something, first take a look at the relevant definition(s):take a look at the relevant definition(s):

x, yx, yA (f(x) = f(y) A (f(x) = f(y) x = y) x = y)

Example:Example:

f:f:RRRR

f(x) = xf(x) = x22

Disproof by counterexample:

f(3) = f(-3), but 3 f(3) = f(-3), but 3 -3, so f is not one-to-one. -3, so f is not one-to-one.

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… … and yet another example:and yet another example:

f:f:RRRR

f(x) = 3xf(x) = 3x

One-to-one: One-to-one: x, yx, yA (f(x) = f(y) A (f(x) = f(y) x = y) x = y)

To show:To show: f(x) f(x) f(y) whenever x f(y) whenever x y y

x x y y 3x 3x 3y 3y f(x) f(x) f(y), f(y),

so if x so if x y, then f(x) y, then f(x) f(y), that is, f is one-to- f(y), that is, f is one-to-one.one.

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A function f:AA function f:AB with A,B B with A,B R is called R is called strictly strictly increasingincreasing, if , if

x,yx,yA (x < y A (x < y f(x) < f(y)), f(x) < f(y)),

and and strictly decreasingstrictly decreasing, if, if

x,yx,yA (x < y A (x < y f(x) > f(y)). f(x) > f(y)).

Obviously, a function that is either strictly Obviously, a function that is either strictly increasing or strictly decreasing is increasing or strictly decreasing is one-to-oneone-to-one..

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A function f:AA function f:AB is called B is called ontoonto, or , or surjectivesurjective, if , if and only if for every element band only if for every element bB there is an B there is an element aelement aA with f(a) = b.A with f(a) = b.

In other words, f is onto if and only if its In other words, f is onto if and only if its rangerange is is its its entire codomainentire codomain..

A function f: AA function f: AB is a B is a one-to-one one-to-one correspondencecorrespondence, or a , or a bijectionbijection, if and only if it is , if and only if it is both one-to-one and onto.both one-to-one and onto.

Obviously, if f is a bijection and A and B are Obviously, if f is a bijection and A and B are finite sets, then |A| = |B|.finite sets, then |A| = |B|.

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Examples:Examples:

In the following examples, we use the arrow In the following examples, we use the arrow representation to illustrate functions f:Arepresentation to illustrate functions f:AB. B.

In each example, the complete sets A and B are In each example, the complete sets A and B are shown.shown.

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Is f injective?Is f injective?

No.No.

Is f surjective?Is f surjective?

No.No.

Is f bijective?Is f bijective?

No.No.

LindaLinda

MaxMax

KathyKathy

PeterPeter

BostonBoston


MoscowMoscow

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No.No.


Yes.Yes.


No.No.

LindaLinda

MaxMax

KathyKathy

PeterPeter

BostonBoston


MoscowMoscow

PaulPaul

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Yes.Yes.


No.No.


No.No.

LindaLinda

MaxMax

KathyKathy

PeterPeter

BostonBoston


MoscowMoscow

LLüübeckbeck

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No! f is not evenNo! f is not evena function!a function!

LindaLinda

MaxMax

KathyKathy

PeterPeter

BostonBoston


MoscowMoscow

LLüübeckbeck

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Yes.Yes.


Yes.Yes.


Yes.Yes.

LindaLinda

MaxMax

KathyKathy

PeterPeter

BostonBoston


MoscowMoscow

LLüübeckbeckHelenaHelena

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InversionInversion

An interesting property of bijections is that An interesting property of bijections is that they have an they have an inverse functioninverse function..

The The inverse functioninverse function of the bijection of the bijection f:Af:AB is the function fB is the function f-1-1:B:BA with A with

ff-1-1(b) = a whenever f(a) = b. (b) = a whenever f(a) = b.

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InversionInversion

Example:Example:




f(Peter) = Lf(Peter) = Lüübeckbeck

f(Helena) = New Yorkf(Helena) = New York

Clearly, f is bijective.Clearly, f is bijective.

The inverse function fThe inverse function f--

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ff-1-1(Moscow) = Linda(Moscow) = Linda

ff-1-1(Boston) = Max(Boston) = Max

ff-1-1(Hong Kong) = Kathy(Hong Kong) = Kathy

ff-1-1(L(Lüübeck) = Peterbeck) = Peter

ff-1-1(New York) = Helena(New York) = Helena

Inversion is only Inversion is only possible for bijectionspossible for bijections(= invertible functions)(= invertible functions)

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InversionInversion

LindaLinda

MaxMax

KathyKathy

PeterPeter

BostonBoston


MoscowMoscow

LLüübeckbeckHelenaHelena

ff

ff-1-1

ff-1-1:C:CP is no P is no function, because function, because it is not defined it is not defined for all elements of for all elements of C and assigns two C and assigns two images to the pre-images to the pre-image New York.image New York.

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CompositionCompositionThe The compositioncomposition of two functions g:A of two functions g:AB B and f:Band f:BC, denoted by fC, denoted by fg, is defined by g, is defined by

(f(fg)(a) = f(g(a))g)(a) = f(g(a))

This means that This means that • firstfirst, function g is applied to element , function g is applied to element aaA,A, mapping it onto an element of B, mapping it onto an element of B,• thenthen, function f is applied to this element , function f is applied to this element of of B, mapping it onto an element of C. B, mapping it onto an element of C.• ThereforeTherefore, the composite function maps , the composite function maps from A to C. from A to C.

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CompositionComposition

Example:Example:

f(x) = 7x – 4, g(x) = 3x,f(x) = 7x – 4, g(x) = 3x,

f:f:RRRR, g:, g:RRRR

(f(fg)(5) = f(g(5)) = f(15) = 105 – 4 = 101g)(5) = f(g(5)) = f(15) = 105 – 4 = 101

(f(fg)(x) = f(g(x)) = f(3x) = 21x - 4g)(x) = f(g(x)) = f(3x) = 21x - 4

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CompositionComposition

Composition of a function and its inverse:Composition of a function and its inverse:

(f(f-1-1f)(x) = ff)(x) = f-1-1(f(x)) = x(f(x)) = x

The composition of a function and its The composition of a function and its inverse is the inverse is the identity functionidentity function i(x) = x. i(x) = x.

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GraphsGraphs

TheThe graphgraph of a functionof a function f:Af:AB is the set of B is the set of ordered pairs {(a, b) | aordered pairs {(a, b) | aA and f(a) = b}.A and f(a) = b}.

The graph is a subset of AThe graph is a subset of AB that can be B that can be used to visualize f in a two-dimensional used to visualize f in a two-dimensional coordinate system.coordinate system.

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Floor and Ceiling FunctionsFloor and Ceiling Functions

The The floorfloor and and ceilingceiling functions map the real functions map the real numbers onto the integers (numbers onto the integers (RRZZ).).

The The floorfloor function assigns to r function assigns to rRR the largest the largest zzZZ with z with z r, denoted by r, denoted by rr..

Examples:Examples: 2.32.3 = 2, = 2, 22 = 2, = 2, 0.50.5 = 0, = 0, -3.5-3.5 = - = -44

The The ceilingceiling function assigns to r function assigns to rRR the the smallest zsmallest zZZ with z with z r, denoted by r, denoted by rr..

Examples:Examples: 2.32.3 = 3, = 3, 22 = 2, = 2, 0.50.5 = 1, = 1, -3.5-3.5 = - = -3 3

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ExercisesExercises

I recommend Exercises 1 and 15 in Section I recommend Exercises 1 and 15 in Section 1.6.1.6.

It may also be useful to study the graph It may also be useful to study the graph displays in that section. displays in that section.

Another question: What do all graph displays Another question: What do all graph displays for any function f:for any function f:RRR R have in common?have in common?

Have a nice weekend! Have a nice weekend!

functions

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