lecture 3 functions

8/8/2019 Lecture 3 Functions

1/20

Lecture 3

Functions

Chapter 2.3

Kenneth Rosen Book


2/20

Functions

Def: Let A and B be sets. A function f (or more

completely, f : A B) is a rule that assigns to

each elementa A exactly one element f(a) B,

called the value off at a.

We also say that f : A B is a mapping from

domain A to codomain B.

f(a) is called the image of the element a, and the

element a is called a preimage of f(a).


3/20

Graphical Representations

Functions can be represented graphically

in several ways:

AB

a b

f

f

x

y

PlotBipartite Graph

Like Venn diagrams

A B


4/20

Some Function Terminology

If it is written that f:ApB, and f(a)=b(where aA &bB), then we say:A is the domain off.

B is the codomain off.

b is the image ofa underf.

a is apre-image ofb underf.

In general,b may have more than 1 pre-image.

The range RB offis R={b | a f(a)=b }.


5/20

Range versus Codomain

The range of a function might notbe its

whole codomain.

The codomain is the set that the functionis declaredto map all domain values into.

The range is theparticularset of values in

the codomain that the function actuallymaps elements of the domain to.


6/20

Range vs. Codomain - Example

Suppose I declare to you that: fis a

function mapping students in this class to

the set of grades {A,B,C,D,E}.

At this point, you know fs codomain is:__________, and its range is ________.

Suppose the grades turn out all As and

Bs. Then the range offis _________, but its

codomain is __________________.

{A,B,C,D,E} unknown!

{A,B}

still {A,B,C,D,E}!


7/20

Function Operator Example

, (plus,times) are binary operatorsoverR. (Normal addition & multiplication.)

Therefore, we can also add and multiplyfunctions f,g:RpR: (f g):RpR, where (f g)(x) = f(x) g(x)

(f g):RpR, where (f g)(x) = f(x) g(x)


8/20

One-to-One Functions

A function is one-to-one (1-1), orinjective,

oran injection, iff every element of its

range has only1 pre-image.

Bipartite (2-part) graph representationsof functions that are (or not) one-to-one:

One-to-one

Not one-to-one

Not even a

function!


9/20

Onto (Surjective) Functions

A function f:ApB is onto orsurjective ora surjection iff its range is equal to its

codomain (bB,aA: f(a)=b).

Think: An onto function maps the setA

onto (over, covering) the entiretyof the

set B, not just over a piece of it.


10/20

Illustration of Onto

Some functions that are, or are not,ontotheir codomains:

Onto(but not 1-1)

Not Onto(or 1-1)

Both 1-1and onto

1-1 butnot onto


11/20

Some more examples


12/20

Bijections

A function fis said to be a one-to-one

correspondence, ora bijection, or

reversible, orinvertible, iff it is

both one-to-one and onto.

For bijections f:ApB, there exists aninverse of f, written f1:BpA, which is theunique function such that

(where IA is the identity function onA)AIff ! Q

1


13/20

Composition of f and g


14/20

Graphs of Functions

We can represent a function f:ApB as aset of ordered pairs {(a,f(a)) | aA}.

Note that a, there is only 1 pair(a,b).

For functions over numbers, we can

represent an ordered pair (x,y) as a point

on a plane. A function is then drawn as a curve (set of

points), with only one yfor eachx.


15/20

A Couple of Key Functions

In discrete math, we will frequently use the

following two functions over real numbers:

The floorfunction -

:RZ, where -x (floor ofx) means the largest (most positive) integerex. I.e.,-x : max({iZ|ix}).

The ceilingfunction :RZ, where x(ceiling ofx) means the smallest (most

negative) integerux. -x : min({iZ|ix})


16/20

Visualizing Floor& Ceiling

Real numbers fall to their floor or rise to

their ceiling.

Note that ifxZ,-x { -x &x { x

Note that ifxZ,-x = x =x.

0

1

1

2

3

2

3

..

.

.

. .

. . .

1.6

1.6=2

-1.4= 2

1.4

1.4= 1

-1.6=1

3

3=-3= 3


17/20

Plots with floor/ceiling

Note that forf(x)=-x, the graph offincludes thepoint (a, 0) for all values ofa such that au0 anda


18/20

Plots with floor/ceiling: Example

Plot of graph of function f(x) = -x/3:

x

f(x)

Set of points (x,f(x))

+3

2

+2

3


19/20

Plots with floor/ceiling


20/20

Review of 2.3 (Functions)

Function variables f,g,h,

Notations: f:ApB, f(a),f(A).

Terms: image, preimage, domain, codomain,

range, one-to-one, onto, strictly (in/de)creasing,

bijective, inverse, composition.

Function unary operatorf1,

binary operators ,,etc., and . The RpZ functions -x and x.

lecture 3 functions

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