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Functions. Functions. Definition : A function f from set A to set B , denoted f : A → B, is an assignment of each element of A to exactly one element of B . We write f ( a ) = b if b is the unique element of B assigned to the element a of A . .

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functions1
Functions
  • Definition: A functionf from set A to set B, denoted f: A→B,is an assignment of each element of A to exactly one element of B.
  • We writef(a) = bif b is the unique element of B assigned to the element a of A.
  • Functions are also called mappings

Students

Grades

A

Carlota Rodriguez

B

Sandeep Patel

C

Jalen Williams

D

F

Kathy Scott

functions2
Functions

Given a function f: A→ B

  • A is called the domain of f
  • B is called the codomain of f
  • f is a mapping from Ato B
  • If f(a) = b
    • then bis called the image of aunder f
    • a is called the preimage of b
  • The range (or image) of f is the set of all images of points in A. We denote it by f(A).
example
Example

A

B

  • The domain of f is A
  • The codomain of f is B
  • The image of b is y
    • f(b) = y
  • The preimage of y is b
  • The preimageof z is {a,c,d}
  • The range/image of A is {y,z}
    • f(A) = {y,z}

a

x

b

y

c

z

d

representing functions
Representing Functions
  • Functions may be specified in different ways:
    • An explicit statement of the assignment.
      • Students and grades example.
    • A formula.
      • f(x) = x+ 1
    • A computer program.
      • A Java program that when given an integer n, produces the nth Fibonacci Number
injections
Injections

B

A

  • Definition: A function f is one-to-one, or injective, iffa ≠ b implies that f(a) ≠ f(b) for all a and b in the domain of f.

x

a

v

b

y

c

z

d

w

surjections
Surjections

A

B

  • Definition: A function f from A to B is called onto or surjective, iff for every element b ∈ B there exists an element a ∈A with f(a) = b

x

a

b

y

c

z

d

bijections
Bijections

A

B

  • Definition: A function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto (surjective and injective)

a

x

b

y

c

z

d

w

showing that f is is not injective or surjective
Showing that f is/is not injective or surjective
  • Consider a function f: A→ B
  • f is injectiveiff:

∀x,y∈ A ( x ≠ y → f(x) ≠ f(y) )

  • f is not injective iff:

∃x,y∈ A( x ≠ y ∧ f(x) = f(y) )

  • fis surjectiveiff:

∀y ∈ B ∃x ∈ A( f(x) = y )

  • f is not surjectiveiff:

∃y ∈ B ∀x ∈ A (f(x) ≠ b)

inverse functions
Inverse Functions
  • Definition: Let f be a bijection from A to B. Then the inverse of f, denoted f –1, is the function from B to Adefined as
  • No inverse exists unless f is a bijection.
inverse functions1
Inverse Functions
  • Example 1:
    • Let f be the function from {a,b,c} to {1,2,3}
    • f(a)=2, f(b)=3, f(c)=1.
    • Is f invertible and if so what is its inverse?
  • Solution:
    • fis invertible because it is a bijection
    • f –1reverses the correspondence given by f:
    • f –1(1)=c, f –1(2)=a, f –1(3)=b.
inverse functions2
Inverse Functions
  • Example 2:
    • Let f: R→Rbe such that f(x) = x2
    • Is f invertible, and if so, what is its inverse?
  • Solution:
    • The function f is not invertible because it is not one-to-one
inverse functions3
Inverse Functions
  • Example 3:
    • Let f: Z  Zbe such that f(x) = x + 1
    • Is f invertible and if so what is its inverse?
  • Solution:
    • The function f is invertible because it is a bijection
    • f –1reverses the correspondence:
    • f –1(y) = y – 1
composition
Composition
  • Definition: Let f: B→C, g: A→B. The composition of f with g, denoted f∘g is the function from A to C defined by
composition1
Composition

A

g

B

f

C

A

C

v

a

h

a

b

i

h

w

c

b

i

j

x

d

c

j

y

d

composition2
Composition
  • Example: If and then

and

graphs of functions
Graphs of Functions
  • Let f be a function from the set A to the set B. The graph of the function f is the set of ordered pairs

{(a,b) | a ∈Aandf(a) = b}

Graph of f(x) = x2

from Z to Z

Graph of f(n) = 2n+1

from Z to Z

some important functions
Some Important Functions
  • The floor function, denoted

is the largest integer less than or equal to x.

  • The ceilingfunction, denoted

is the smallest integer greater than or equal to x

  • Examples:
some important functions1
Some Important Functions

Floor Function (≤x) Ceiling Function (≥x)

factorial function
Factorial Function
  • Definition: f: N → Z+, denoted by f(n) = n! is the product of the first n positive integers:

f(n) = 1 ∙ 2 ∙∙∙ (n–1) ∙ nforn>0

f(0) = 0! = 1

  • Examples:
    • f(1) = 1! = 1
    • f(2) = 2! = 1 ∙ 2 = 2
    • f(6) = 6! = 1 ∙ 2 ∙ 3 ∙ 4 ∙ 5 ∙ 6 = 720
    • f(20) = 2,432,902,008,176,640,000
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