3.4
The Function Relation

3.4.1 Introduction

No doubt the concept of a function covers familiar territory for many readers of this book.1 Normally, in the beginning of mathematics books, a function f : A → B is defined as a rule that assigns to each value x ∈ A a unique value y ∈ B. This is the definition proposed by German mathematician Peter Lejeune Dirichlet (1805–1859) in the 1830s. When we write an algebraic formula like

equation

where x is taken as a real number, the rule is clearly understood, it assigns to each x the value sin x. We denote the function by the letter f and the value of the function at x by f(x). This motivates the Dirichlet definition of a function.

Sketch of Peter Gustav Lejeune Dirichlet.
Peter Gustav Lejeune Dirichlet (1805–1859)
Diagram of a function displaying a box labeled A=domain(f) with solid circle for x linked by an arrow to a solid circle for f(x) inside a box enclosed by another bigger box labeled B=codomain(f).

Figure 3.27 Illustration of a function.

There are several synonyms for the word “function.” The words mapping (or map), transformation, and operator are often used depending on the context as well as the domain and codomain of the function.

3.4.2 Relation Definition of a Function

In addition to defining a function as a rule (ala Dirichlet), we can also think of a function in terms of relations.

3.4.3 Composition of Functions

The function f(x) = sin x2 can be interpreted as assigning x → sin x2. However, it can also be interpreted as a combination or composition of two functions: the first assigning x → x2, the second assigning x2 → sin x2, which leads us to the following definition.

Four graphs of functions ℝ → ℝ displaying curves for y=x2 (upper left), y=x3– x (upper right), y=ex (lower left), and y=x3 (lower right).

Figure 3.30 Types of functions ℝ → ℝ.

Table 3.6 An injection.

n 1 2 3 4 n
f(n) = n2 1 4 9 16 n2

3.4.4 Inverse Functions

In arithmetic, some numbers have inverses. For example −3 is the additive inverse of +3 since 3 + (−3) = 0. Some functions also have inverses in the sense that the inverse “undoes” the operation of the function.

Diagram of inverse function displaying an ellipse labeled x with right arrow labeled f and left arrow labeled f –1 linking to an ellipse labeled y enclosed by another bigger ellipse.

Figure 3.31 Inverse function.

Some common inverses of 1–1 functions defined of given domains are listed in Table 3.7.

Problems

  1. Testing Relations

    Determine which of the following relations are functions. For functions, what is the domain and range of the function?

    1. R = {(1, 3), (3, 4), (4, 1), (2, 1)}
    2. R = {(1, 3), (1, 4), (1, 2), (3, 1)}
    3. R = {(1, 3), (3, 4), (1, 1)}
    4. R = {(1, 2), (2, 2), (3, 2), (2, 3)}
  2. Graphing Relations and Functions

    Graph each of the following relations on ℝ and tell which relations are functions.

    1. R = {(x, y) : y = x2}
    2. images
    3. images
    4. R = {(x, y) : x = |y|}
    5. R = {(x, y) : |x| + |y| = 1}
  3. Find the Mystery Function

    Find a function that “tears” the interval [0, 1] into two parts at its midpoint and then “stretches” each part uniformly to twice its length.

  4. Compositions

    Find f ∘ g and g ∘ f and their domains for the following functions f and g. We assume the domains of the functions are all values for which the function is well defined.

    1. images
    2. f(x) = 2x + 3, g(x) =  − x2 + 5
    3. images
    4. f(x) = |x|, g(x) = |x|
    5. images
    6. images
  5. Composition of Three Functions

    For each function f, g, h below that maps {1, 2, 3, 4} to itself, find the composition f ∘ (g ∘ h).

    equation
  6. Backwards Compositions

    One can sometimes interpret a function h as a composition of two functions. For the given function h given below determine f, g such that h = f ∘ g.

    1. h(x) = (x − 1)2 + (x − 1) + 3
    2. h(x) = sin(1/x)
    3. h(x) = x2 + x + 1
    4. images
  7. Decomposing a Function as a Composition

    Write the function h(x) = x2 + 1 as a composition h = f ∘ g of two functions in an infinite number of different ways.

  8. Classroom Function

    Let A be the set of students in your Intro to Abstract Math Class and B be the natural numbers from 1 to 100.

    1. To each student, assign the student's age. Is this a function from A to B?
    2. To each natural number n ∈ B, assign students in the class whose age is n. Is this a function from B to A?
  9. More Compositions

    Given functions f, g illustrated in Figure 3.33, both having domains and codomains A = {1, 2, 3, 4}, find the following.

    1. f ∘ g
    2. g ∘ f
    3. f ∘ f
    4. g ∘ g
    Graphs of x and y displaying solid circle markers on horizontal points 1, 2, 3, and 4 and vertical points 1, 2, and 4 (left) and horizontal points 1, 2, 3, and 4 and vertical points 2, 3, and 4 (right).

    Figure 3.33 Compositions.

  10. Shifting Domain of a Composition

    Given the function defined by

    equation

    whose domain is the real numbers, except 1, find the domain of f ∘ f.

  11. Graphing a Composition

    Draw the graph for two arbitrary real‐valued functions f, g of a real variable. Then select an arbitrary real number x and use the graphs to find the location of (f ∘ g)(x).

  12. Compositions

    Find f ∘ g if

    • f : ℝ → ℝ3, f(t) = (t, t2, t3)
    • g : ℝ → ℝ, g(t) = sin t
  13. Composition of Operators

    Given the differential operators

    • L1( f ) = x f(x) + 1
    • images

    find

    1. L1 ∘ L2
    2. L2 ∘ L1
  14. Recursive Functions

    A recursive function is one that is defined in terms of itself, normally defined over a restricted subset of its domain. For example, the factorial function n !  = n(n − 1)(n − 2)⋯(2)(1) can be defined recursively as

    equation

    Another example of a recursively defined function is the greatest common divisor of two positive integers m and n, which is defined as the largest positive integer that divides both m and n. For 0 < n ≤ m, we can define the greatest common divisor of m and n by

    equation

    Use this recursive definition to find the greatest common divisor of the following numbers.

    1. m = 25, n = 5
    2. m = 101, n = 13
    3. m = 37, n = 3
  15. Functional Equation

    A functional equation is an equation which expresses the value of the function at a point in terms of the value of the function at another point or points. Below, are listed four well‐known functional equations. Find a function or functions that satisfies the given functional equation.

    1. f(x + y) = f(x) + f(y)
    2. f(x + y) = f(x)f(y)
    3. f(xy) = f(x) + f(y), x, y > 0
    4. f(xy) = f(x)f(y), x, y > 0
  16. Injections, Surjections, Bijections

    Give examples of the following functions f1, f2, f3, f4 from ℕ to ℕ that satisfy the following properties.

    1. f1 is neither 1–1 or onto.
    2. f2 is 1–1, but not onto.
    3. f3 is onto, but not 1–1.
    4. f4 is both 1–1 and onto.
  17. Find the Function

    Find a function f that satisfies the following properties.

    1. f maps ℝ to {1, 2, 3}
    2. f maps ℕ to ℝ
    3. f maps ℝ × ℝ to ℝ
    4. f maps ℝ to ℝ × ℝ
    5. f maps {a, b, c} to [0, 1]
  18. Injections, Surjections, and Bijections

    Which of the following functions f : ℝ → ℝ are injective, surjective, bijective, or none of the three. Assume the domains of the functions are subsets of ℝ for which the function is well‐defined.

    1. f(x) = x3 − 2x + 1
    2. f(x) = sin(1/x)
    3. images
    4. f(x) = e−x
  19. Interesting Function

    Let f : ℕ → ℕ be the function defined by

    • even numbers 2n → n
    • odd numbers 2n − 1 → n
    1. Draw part of the graph of this function.
    2. Is this function 1–1?
    3. Is the function an onto function?
  20. Inverse Function

    Given the function defined by

    equation
    1. Draw the graph of f
    2. Find the domain and range of f.
    3. Prove that the function is 1–1.
    4. Find the inverse of the function.
    5. Find the domain and range of the inverse function.
    6. Draw the graph of the inverse function.
  21. Function as Ordered Pairs

    For f : {1, 2, 3} → ℕ defined by the ordered pairs f = {(1, 3), (2, 5), (3, 1)}:

    1. Is f 1–1?
    2. Is f onto?
    3. What is the range of f ?
  22. 1–1 But Not Onto

    Give an example of a function f : ℝ → ℝ that is 1–1, but not onto.

  23. Hmmmmmmmmm

    For what value of the exponent n ∈ ℕ is the function f(x) = xn a 1–1 function?

  24. Counting Functions I

    Let A = {1, 2}, B = {a, b, c}. Answer the following questions. Hint: It may help by drawing a simple picture.

    1. How many functions are there from A to B?
    2. How many 1–1 functions are there from A to B?
    3. How many onto functions are there from A to B?
    4. How many one‐to‐one correspondences are there from A to B?
  25. Counting Functions II

    Let A = {1, 2, 3}, B = {a, b}. Answer the following questions. Hint: It may help by drawing a simple picture.

    1. How many functions are there from A to B?
    2. How many 1–1 functions are there from A to B?
    3. How many onto functions are there from A to B?
    4. How many one‐to‐one functions are there from A to B?
  26. Finding Injections and Surjections
    1. Find a function f : ℕ → ℕ that is 1–1 but not onto.
    2. Find a function f : ℕ → ℕ that is onto but not 1–1.
  27. Composition of Onto Maps

    Prove that if

    equation

    then the composition f ∘ g is an onto function from X to Z. In short, the composition of surjections is a surjection.

  28. Composition of 1–1 Functions

    Prove that if g is a 1–1 mapping from X to Y, and f is a 1–1 mapping from Y to Z, then the composition f ∘ g is a 1–1 mapping from X to Z. In other words, the composition of injections is an injection.

  29. Hmmmmmmmmmm

    If

    equation

    Find examples of the following:

    1. A 1–1 composition f ∘ g, where f is not 1–1.
    2. An onto composition f ∘ g : A → C, where g : A → B is not onto.
    3. A one‐to‐one correspondence f ∘ g, where g is not onto and f is onto.
  30. More Counting Functions

    Let S = {1, 2, 3}.

    1. How many functions are there from S to S?
    2. How many onto functions are there from S to S?
    3. How many 1–1 functions are there from S to S?
    4. How many bijections are there from S to S?
  31. Counting Functions in General

    If a set A has m elements and B has n elements, how many functions of different types map A into B?

    1. All functions
    2. All 1–1 functions
    3. All bijections
  32. Euler Totient Function

    In number theory, the Euler totient function ϕ(n) (or phi function) is a function

    equation

    defined on the natural numbers that gives the number of natural numbers less than n that are coprime with n, where a number is coprime with another if the greatest common divisor of the two numbers is 1. For example ϕ(6) = 2 since 1 and 5 are coprime with 6, but 2, 3, and 4 are not. Verify the following special cases of some important properties of the Euler totient function.

    1. ϕ(17) = 16, general theorem ϕ(p) = p − 1, p prime
    2. images
    3. ϕ(15) = ϕ(3)ϕ(5), general theorem ϕ(mn) = ϕ(m)ϕ(n), m, n coprime
    4. ϕ(53) = (5 − 1)52, general theorem ϕ(pk) = (p − 1)pk − 1
  33. Carmichael's Totient Function Conjecture

    An open question in number theory is the Carmichael Totient Function Conjecture, which states that for every natural number n, there is at least one other natural number m that satisfies ϕ(m) = ϕ(n). In other words, both have the same number of coprimes. As of 2019, it is unknown whether the conjecture is true or false.

  34. Internet Research

    There is a wealth of information related to topics introduced in this section just waiting for curious minds. Try aiming your favorite search engine toward strange functions, history of the function, and list of mathematical functions.

Notes

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