Functions

John

The following are some notes taken during a lecture.

Basic Definitions

Definition: Let A and B be nonempty sets. A function f  from A to 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 by the function f to the element a of A. 

Functions are sometimes called mappings or transformations.

 

A function f: A → B  can also be defined as a subset of A×B (a relation). This subset is restricted to be a relation where no two elements of the relation have the same first element. 

Specifically, a function f from A to B contains one, and only one ordered pair (a, b) for every element a∈ A.

 

Basic Definitions

  • Given a function f: A → B: 
  • We say f maps A to B or f is a mapping from  A to B
  • A is called the domain of f
  • B is called the codomain of f
  • If f(a) = b, then b is called the image of a under f. a is called the preimage of b.
  • The range of f is the set of all images of points in A under f. We denote it by f(A).
  • Two functions are equal when they have the same domain, the same codomain and map each element of the domain to the same element of the codomain. 

Representing Functions

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

 

Examples

Injections

Definition: A function f is said to be one-to-one ,  or injective, if and only if f(a) = f(b) implies that  a = b for all a and b in the domain of f. A function is said to be an injection if it is one-to-one.

 

Surjection

Definition: A function f from A to B is called onto or surjective, if and only if for every element there is an element with. A function f is called a surjection if it is onto.

Bijection

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).

 

Showing that a Function is One-to-One or Onto

 

Showing that a Set is One-to-One or Onto

Example 1: Let f be the function from {a,b,c,d} to {1,2,3} defined by f(a) = 3, f(b) = 2, f(c) = 1, and f(d) = 3. Is f an onto function?

Solution: Yes, f is onto since all three elements of the codomain are images of elements in the domain. If the codomain were changed to {1,2,3,4}, f  would not be onto. 

Example 2: Is the function  f(x) = x2    from the set of integers to the set of integers onto?  

Solution: No, f is  not onto because there is no integer x with x2  = −1, for example. 

 

Inverse Functions

Definition: Let f be a bijection from A to B. Then the inverse of f, denoted          , is the function from B to A defined as No inverse exists unless f is a bijection. Why?

 

 

Examples

Example 1: Let f be the function from {a,b,c} to {1,2,3} such that f(a) = 2, f(b) = 3, and f(c) = 1. Is f invertible and if so what is its inverse?

Solution: The function f is invertible because it is a one-to-one correspondence. The inverse function f-1  reverses the correspondence given by f, so f–1 (1) = c,    f–1 (2) = a,  and f–1 (3) = b.

 

Example 2: Let f: Z 🡪 Z be 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 one-to-one correspondence. The inverse function f-1  reverses the correspondence  so f–1 (y) = y – 1.   

 

Example 3: Let f: R → R be such that                    .    Is f invertible, and if so, what is its inverse? 

Solution: The function f is not invertible because it is not one-to-one . 

 

Composition of Functions

Definition: Let f: B → C, g: A → B. The composition of f with g, denoted            is the function from A to C defined by


 

Example 2: Let g be the function from the set {a,b,c} to itself such that g(a) = b, g(b) = c, and g(c) = a. Let  f be the function from the set {a,b,c} to the set {1,2,3} such that     f(a) = 3, f(b) = 2, and f(c) = 1.

    What is the composition of f and g, and what is the composition of g and f.

Solution:  The composition f∘g  is defined by 

  • f∘g  (a)= f(g(a)) = f(b) = 2.
  • f∘g  (b)= f(g(b)) = f(c) = 1.
  • f∘g  (c)= f(g(c)) = f(a) = 3. 

Note that g∘f  is not defined, because the range of f is not a subset of the domain of g. 

 

Example 2: Let f and g be functions from the set of integers to the set of integers defined by  f(x) = 2x + 3 and g(x) = 3x + 2. 

What is the composition of f and g, and also the composition of g and f ?

Solution:

  • f∘g  (x)= f(g(x)) = f(3x + 2) = 2(3x + 2) + 3 = 6x + 7
  • g∘f  (x)= g(f(x)) = g(2x + 3) = 3(2x + 3) + 2 = 6x + 11 

 

Graphs of Functions

 

Some Important Functions

Floor Function

Floor and Ceiling Function

Proving Properties of Functions

 

Factorial Functions

Definition:  f: N → Z+ , denoted by f(n) = n! is the product of the first n positive integers when n is a nonnegative integer.

        f(n) = 1 ∙ 2 ∙∙∙ (n – 1) ∙ n,         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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