Functions

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# Functions - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Functions' - jaeger

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### Functions

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

A

Carlota Rodriguez

B

Sandeep Patel

C

Jalen Williams

D

F

Kathy Scott

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

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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}

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

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• 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.

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Surjections

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• 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

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Bijections

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• 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)

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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
• 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 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 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 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
• 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
Composition

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Composition
• Example: If and then

and

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
• 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 Functions

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

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