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
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.
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.
In addition to defining a function as a rule (ala Dirichlet), we can also think of a function in terms of relations.
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.
Table 3.6 An injection.
n | 1 | 2 | 3 | 4 | ⋯ | n | ⋯ |
f(n) = n2 | 1 | 4 | 9 | 16 | ⋯ | n2 | ⋯ |
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.
Some common inverses of 1–1 functions defined of given domains are listed in Table 3.7.
Determine which of the following relations are functions. For functions, what is the domain and range of the function?
Graph each of the following relations on ℝ and tell which relations are functions.
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.
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.
For each function f, g, h below that maps {1, 2, 3, 4} to itself, find the composition f ∘ (g ∘ h).
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.
Write the function h(x) = x2 + 1 as a composition h = f ∘ g of two functions in an infinite number of different ways.
Let A be the set of students in your Intro to Abstract Math Class and B be the natural numbers from 1 to 100.
Given functions f, g illustrated in Figure 3.33, both having domains and codomains A = {1, 2, 3, 4}, find the following.
Given the function defined by
whose domain is the real numbers, except 1, find the domain of f ∘ f.
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).
Find f ∘ g if
Given the differential operators
find
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
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
Use this recursive definition to find the greatest common divisor of the following numbers.
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.
Give examples of the following functions f1, f2, f3, f4 from ℕ to ℕ that satisfy the following properties.
Find a function f that satisfies the following properties.
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.
Let f : ℕ → ℕ be the function defined by
Given the function defined by
For f : {1, 2, 3} → ℕ defined by the ordered pairs f = {(1, 3), (2, 5), (3, 1)}:
Give an example of a function f : ℝ → ℝ that is 1–1, but not onto.
For what value of the exponent n ∈ ℕ is the function f(x) = xn a 1–1 function?
Let A = {1, 2}, B = {a, b, c}. Answer the following questions. Hint: It may help by drawing a simple picture.
Let A = {1, 2, 3}, B = {a, b}. Answer the following questions. Hint: It may help by drawing a simple picture.
Prove that if
then the composition f ∘ g is an onto function from X to Z. In short, the composition of surjections is a surjection.
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.
If
Find examples of the following:
Let S = {1, 2, 3}.
If a set A has m elements and B has n elements, how many functions of different types map A into B?
In number theory, the Euler totient function ϕ(n) (or phi function) is a function
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.
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.
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.
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