Functions: Basics and Examples

Functions

Functions

A 

A 

function

function

 f from a set A to a set B is an

 f from a set A to a set B is an

assignment

assignment

 of exactly one element of B to each

 of exactly one element of B to each

element of A.

element of A.

We write

We write

f(a) = b

f(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 write

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

f: A

f: A





B

B

(note:  Here, “

(note:  Here, “





“ has nothing to do with if… then)

“ has nothing to do with if… then)

If f:A

If f:A





B, we say that A is the 

B, we say that A is the 

domain

domain

 of f and B

 of f and B

is the 

is the 

codomain

codomain

 of f.

 of f.

If f(a) = b, we say that b is the 

If f(a) = b, we say that b is the 

image

image

 of a and a is

 of a and a is

the 

the 

pre-image

pre-image

 of b.

 of b.

The 

The 

range

range

 of f:A

 of f:A





B 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:A

We say that f:A





B 

B 

maps

maps

 A to B.

 A to B.

Let us take a look at the function f:P

Let us take a look at the function f:P





C with

C with

P = {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) = Moscow

f(Linda) = Moscow

f(Max) = Boston

f(Max) = Boston

f(Kathy) = Hong Kong

f(Kathy) = Hong Kong

f(Peter) = New York

f(Peter) = New York

Here, the range of f is C.

Here, the range of f is C.

Let us re-specify f as follows:

Let us re-specify f as follows:

f(Linda) = Moscow

f(Linda) = Moscow

f(Max) = Boston

f(Max) = Boston

f(Kathy) = Hong Kong

f(Kathy) = Hong Kong

f(Peter) = Boston

f(Peter) = Boston

Is f still a function?

Is f still a function?

yes

yes

{Moscow, Boston, Hong Kong}

{Moscow, Boston, Hong Kong}

What is its range?

What is its range?

Other ways to represent f:

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 

formula

formula

, e.g.:

, e.g.:

f:

f:

R

R





R

R

f(x) = 2x

f(x) = 2x

This leads to:

This leads to:

f(1) = 2

f(1) = 2

f(3) = 6

f(3) = 6

f(-3) = -6

f(-3) = -6

…

…

Let f

Let f

1

1

 and f

 and f

2

2

 be functions from A to 

 be functions from A to 

R

R

.

.

Then the 

Then the 

sum

sum

 and the 

 and the 

product

product

 of f

 of f

1

1

 and f

 and f

2

2

 are

 are

also functions from A to 

also functions from A to 

R

R

 defined by:

 defined by:

(f

(f

1

1

 + f

 + f

2

2

)(x) =  f

)(x) =  f

1

1

(x) + f

(x) + f

2

2

(x)

(x)

(f

(f

1

1

f

f

2

2

)(x) =  f

)(x) =  f

1

1

(x) f

(x) f

2

2

(x)

(x)

Example:

Example:

f

f

1

1

(x) = 3x,  f

(x) = 3x,  f

2

2

(x) = x + 5

(x) = x + 5

(f

(f

1

1

 + f

 + f

2

2

)(x) =  f

)(x) =  f

1

1

(x) + f

(x) + f

2

2

(x) = 3x + x + 5 = 4x + 5

(x) = 3x + x + 5 = 4x + 5

(f

(f

1

1

f

f

2

2

)(x) =  f

)(x) =  f

1

1

(x) f

(x) f

2

2

(x) = 3x (x + 5) = 3x

(x) = 3x (x + 5) = 3x

2

2

 + 15x

 + 15x

We already know that the 

We already know that the 

range

range

 of a function

 of a function

f:A

f:A





B is the set of all images of elements a

B is the set of all images of elements a





A.

A.

If we only regard a 

If we only regard a 

subset

subset

 S

 S





A, the set of all

A, the set of all

images of elements s

images of elements s





S is called the 

S is called the 

image

image

 of S.

 of S.

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

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

f(S) = {f(s) | s

f(S) = {f(s) | s





S}

S}

Let us look at the following well-known function:

Let us look at the following well-known function:

f(Linda) = Moscow

f(Linda) = Moscow

f(Max) = Boston

f(Max) = Boston

f(Kathy) = Hong Kong

f(Kathy) = Hong Kong

f(Peter) = Boston

f(Peter) = Boston

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}

A function f:A

A function f:A





B is said to be 

B is said to be 

one-to-one

one-to-one

 (or

 (or

injective

injective

), if and only if

), if and only if





x, y

x, y





A (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 the

does not map two distinct elements of A onto the

same element of B.

same element of B.

And again…

And again…

f(Linda) = Moscow

f(Linda) = Moscow

f(Max) = Boston

f(Max) = Boston

f(Kathy) = Hong Kong

f(Kathy) = Hong Kong

f(Peter) = Boston

f(Peter) = Boston

Is f one-to-one?

Is f one-to-one?

No, Max and Peter are

No, Max and Peter are

mapped onto the same

mapped onto the same

element of the image.

element of the image.

g(Linda) = Moscow

g(Linda) = Moscow

g(Max) = Boston

g(Max) = Boston

g(Kathy) = Hong Kong

g(Kathy) = Hong Kong

g(Peter) = New York

g(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.

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

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

x, y





A (f(x) = f(y) 

A (f(x) = f(y) 





 x = y)

 x = y)

Example:

Example:

f:

f:

R

R





R

R

f(x) = x

f(x) = x

2

2

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.

… and yet another example:

… and yet another example:

f:

f:

R

R





R

R

f(x) = 3x

f(x) = 3x

One-to-one: 

One-to-one: 





x, y

x, y





A (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-one.

 f(y), that is, f is one-to-one.

A function f:A

A function f:A





B with A,B 

B with A,B 





 R is called 

 R is called 

strictly

strictly

increasing

increasing

, if

, if





x,y

x,y





A (x < y 

A (x < y 





 f(x) < f(y)),

 f(x) < f(y)),

and 

and 

strictly decreasing

strictly decreasing

, if

, if





x,y

x,y





A (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-one

one-to-one

.

.

A function f:A

A function f:A





B is called 

B is called 

onto

onto

, or 

, or 

surjective

surjective

, if

, if

and only if for every element b

and only if for every element b





B there is an

B there is an

element a

element a





A 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 

range

range

 is

 is

its 

its 

entire codomain

entire codomain

.

.

A function f: A

A function f: A





B is a 

B is a 

one-to-one correspondence

one-to-one correspondence

,

,

or a 

or a 

bijection

bijection

, if and only if it is both one-to-one

, if and only if it is both one-to-one

and onto.

and onto.

Obviously, if f is a bijection and A and B are finite

Obviously, if f is a bijection and A and B are finite

sets, then |A| = |B|.

sets, then |A| = |B|.

Examples:

Examples:

In the following examples, we use the arrow

In the following examples, we use the arrow

representation to illustrate functions f:A

representation to illustrate functions f:A





B.

B.

In each example, the complete sets A and B are

In each example, the complete sets A and B are

shown.

shown.

Is f injective?

Is f injective?

No.

No.

Is f surjective?

Is f surjective?

No.

No.

Is f bijective?

Is f bijective?

No.

No.

Is f injective?

Is f injective?

No.

No.

Is f surjective?

Is f surjective?

Yes.

Yes.

Is f bijective?

Is f bijective?

No.

No.

Is f injective?

Is f injective?

Yes.

Yes.

Is f surjective?

Is f surjective?

No.

No.

Is f bijective?

Is f bijective?

No.

No.

Is f injective?

Is f injective?

No! f is not even

No! f is not even

a function!

a function!

Is f injective?

Is f injective?

Yes.

Yes.

Is f surjective?

Is f surjective?

Yes.

Yes.

Is f bijective?

Is f bijective?

Yes.

Yes.

An interesting property of bijections is that

An interesting property of bijections is that

they have an 

they have an 

inverse function

inverse function

.

.

The 

The 

inverse function

inverse function

 of the bijection f:A

 of the bijection f:A





B

B

is the function f

is the function f

-1

-1

:B

:B





A with

A with

f

f

-1

-1

(b) = a whenever f(a) = b.

(b) = a whenever f(a) = b.

Example:

Example:

f(Linda) = Moscow

f(Linda) = Moscow

f(Max) = Boston

f(Max) = Boston

f(Kathy) = Hong Kong

f(Kathy) = Hong Kong

f(Peter) = L

f(Peter) = L

ü

ü

beck

beck

f(Helena) = New York

f(Helena) = New York

Clearly, f is bijective.

Clearly, f is bijective.

The inverse function

The inverse function

f

f

-1

-1

 is given by:

 is given by:

f

f

-1

-1

(Moscow) = Linda

(Moscow) = Linda

f

f

-1

-1

(Boston) = Max

(Boston) = Max

f

f

-1

-1

(Hong Kong) = Kathy

(Hong Kong) = Kathy

f

f

-1

-1

(L

(L

ü

ü

beck) = Peter

beck) = Peter

f

f

-1

-1

(New York) = Helena

(New York) = Helena

Inversion is only

Inversion is only

possible for bijections

possible for bijections

(= invertible functions)

(= invertible functions)

f

f

-1

-1

:C

:C





P 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.

The 

The 

composition

composition

 of two functions g:A

 of two functions g:A





B and

B and

f:B

f:B





C, denoted by  f

C, denoted by  f





g, is defined by

g, is defined by

(f

(f





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

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

This means that

This means that

•

first

first

, function g is applied to element a

, function g is applied to element a





A,

A,

   mapping it onto an element of B,

   mapping it onto an element of B,

•

then

then

, function f is applied to this element of

, function f is applied to this element of

   B, mapping it onto an element of C.

   B, mapping it onto an element of C.

•

Therefore

Therefore

, the composite function maps

, the composite function maps

   from A to C.

   from A to C.

Example:

Example:

f(x) = 7x – 4, g(x) = 3x,

f(x) = 7x – 4, g(x) = 3x,

f:

f:

R

R





R

R

, g:

, g:

R

R





R

R

(f

(f





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

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

(f

(f





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

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

Composition of a function and its inverse:

Composition of a function and its inverse:

(f

(f

-1

-1





f)(x) = f

f)(x) = f

-1

-1

(f(x)) = x

(f(x)) = x

The composition of a function and its inverse

The composition of a function and its inverse

is the 

is the 

identity function

identity function

 i(x) = x.

 i(x) = x.

The

The

graph

graph

of a function

of a function

f:A

f:A





B is the set of

B is the set of

ordered pairs {(a, b) | a

ordered pairs {(a, b) | a





A and f(a) = b}.

A and f(a) = b}.

The graph is a subset of A

The graph is a subset of A





B that can be used

B that can be used

to visualize f in a two-dimensional coordinate

to visualize f in a two-dimensional coordinate

system.

system.

The 

The 

floor

floor

 and 

 and 

ceiling

ceiling

 functions map the real

 functions map the real

numbers onto the integers (

numbers onto the integers (

R

R





Z

Z

).

).

The 

The 

floor

floor

 function assigns to r

 function assigns to r





R

R

 the largest

 the largest

z

z





Z

Z

 with z 

 with z 





 r, denoted by 

 r, denoted by 





r

r





.

.

Examples:

Examples:





2.3

2.3





 = 2, 

 = 2, 





2

2





 = 2, 

 = 2, 





0.5

0.5





 = 0, 

 = 0, 





-3.5

-3.5





 = -4

 = -4

The 

The 

ceiling

ceiling

 function assigns to r

 function assigns to r





R

R

 the smallest

 the smallest

z

z





Z

Z

 with z 

 with z 





 r, denoted by 

 r, denoted by 





r

r





.

.

Examples:

Examples:





2.3

2.3





 = 3, 

 = 3, 





2

2





 = 2, 

 = 2, 





0.5

0.5





 = 1, 

 = 1, 





-3.5

-3.5





 = -3

 = -3