Functions

Definition of the Domain of a Function

If f is a function from A to B, we say that A is the *domain* of f.

Definition of the Codomain of a Function

If f is a function from A to B, we say that B is the *codomain* of f. Remarks: (1) The *codomain* is the set of values that *could possibly* come out; sometimes we do not know the exact range of a function (because the function may be complicated or not fully known), but we know the set it lies in (such as the set of all integers). So we define the codomain to address this. (2) We can define (or restrict) the codomain of a function, but not the range. What we choose for the codomain can actually affect whether something is a function or not.

Definition of the Image and Preimage of a Function

If f(a) = b, we say that b is the *image* of a and a is a *preimage* of b.

When we define a function we specify...

(1) Its domain (2) Its codomain (3) The mapping of elements of the domain to elements in the codomain

Functions that never assign the same value to two different domain elements are said to be...

*one-to-one* functions. In other words, if f : A → B, then f being a one-to-one function means that every element of B is the image of a *unique* element of A.

Functions for which every member of the codomain is the image of some element of the domain are called...

*onto* functions. In other words, if f : A → B, then f being an onto function means that every element of B is the image of some element in A.

To show that a function f is injective...

(1) Show that if f(x) = f(y) for arbitrary x, y ∈ A with x ≠ y, then x = y, that is, show that ∀(x ∈ A)∀(y ∈ A)(x ≠ y → f(x) ≠ f(y)) (2) Show that it is strictly increasing or strictly decreasing

Two real-valued functions (and hence two integer-valued functions) with the same domain can be...

... added, as well as multiplied.

If both f and g are one-to-one functions, then f ◦ g is...

... also one-to-one.

When the composition of a function and its inverse is formed, in either order...

... an identity function is obtained. That is, for any invertible function f: A → B, f^(− 1) ◦ f = ι_A f ◦ f ^(− 1) = ι_B, where ι_A and ι_B are the identity functions on the sets A and B, respectively. That is, (f^(− 1))^(− 1) = f

If a function f is not a one-to-one correspondence, we cannot...

... define an inverse function of f. For example, consider the function f: R → R such that f(x) = x^2. Notice that we cannot discern which x is the output, starting from y (if y = 4, x might be -2 or 2). In fact, from the other end, the result is *not* a function, as, it would have two outputs corresponding to a given input on many occasions.

A function is called integer-valued if...

... its codomain is the set of integers.

A function is called real-valued if...

... its codomain is the set of real numbers.

Definition of a Partial Function

A *partial function* f from a set A to a set B is an assignment to each element a in a subset of A, called the *domain of definition* of f, of a unique element b in B. The sets A and B are called the *domain* and *codomain* of f, respectively. We say that f is *undefined* for elements in A that are not in the domain of definition of f . When the domain of definition of f equals A, we say that f is a *total function*. Remarks: (1) We write f : A → B to denote that f is a partial function from A to B. Note that this is the same notation as is used for functions. The context in which the notation is used determines whether f is a partial function or a total function. Example: The function f : Z → R where f (n) = √(n) is a partial function from Z to R where the domain of definition is the set of nonnegative integers. Note that f is undefined for negative integers

Definition of an Onto Function

A function f from A to B is called *onto*, or a *surjection*, if and only if for every element b ∈ B there is an element a ∈ A with f (a) = b. A function f is called surjective if it is onto. A function f is onto if ∀y∃x(f(x) = y), where the domain for x is the domain of the function and the domain for y is the codomain of the function.

Definition of a One-to-One Function

A function f is said to be *one-to-one*, or an *injunction*, 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 *injective* if it is one-to-one. Remarks: (1) By the contrapositive, a function f is one-to one if and only if f(a) ≠ f(b) whenever a ≠ b. (2) We can express that f is one-to-one using quantifiers as ∀a∀b(f(a) = f(b) → a = b) or equivalently ∀a∀b(a ≠ b → f(a) ≠ f(b)), where the domain of discourse is the domain of the function.

Definition of a Function that is Decreasing

A function f whose domain and codomain are subsets of the set of real numbers is called *decreasing* if f(x) ≥ f(y) whenever x < y and x and y are in the domain of f, that is, a function f is decreasing if ∀x∀y(x < y → f(x) ≥ f(y))

Definition of a Function that is Increasing

A function f whose domain and codomain are subsets of the set of real numbers is called *increasing* if f(x) ≤ f(y) whenever x < y and x and y are in the domain of f, that is, a function f is increasing if ∀x∀y(x < y → f(x) ≤ f(y))

Definition of a Function that is Strictly Increasing

A function f whose domain and codomain are subsets of the set of real numbers is called *strictly increasing* if f(x) < f(y), whenever x < y and x and y are in the domain of f, that is, a function f is strictly increasing if ∀x∀y(x < y → f(x) < f(y))

Definition of a Function that is Strictly Decreasing

A function f whose domain and codomain are subsets of the set of real numbers is called strictly increasing if f(x) > f(y), whenever x < y and x and y are in the domain of f, that is, a function f is strictly decreasing if ∀x∀y(x < y → f(x) > f(y))

Definition of an Non-Invertible Function

A function is *not invertible* if it is not a one-to one correspondence, because the inverse of such a function does not exist.

Conditions that guarantee a function is one-to-one:

A function that is either *strictly increasing* or *strictly decreasing* must be one-to-one. However, a function that is increasing, but not strictly increasing, or decreasing, but not strictly decreasing, is *not* one-to-one.

Definition of an Invertible Function

A one-to-one correspondence function is called *invertible* because we can define an inverse of this function.

To show that a function f is surjective...

Consider an arbitrary element y ∈ B and find an element x ∈ A such that f(x) = y. ∀(y ∈ B)∃(x ∈ A)(f(x) = y)

To show that a function f is *not* surjective...

Find a particular y ∈ B such that f(x) ≠ y for all x ∈ A, that is, show that ∀(x ∈ A)∃(y ∈ B)(f(x) ≠ y)

To show that a function f is *not* injective...

Find particular elements x, y ∈ A such that x ≠ y and f(x) = f(y), that is, show that ∃(x ∈ A)∃(y ∈ A)(x ≠ y ∧ f(x) = f(y))

Useful Properties of the Floor and Ceiling Functions

If n is an integer and x is a real number, then (1a) floor(x) = n if and only if n ≤ x < n + 1 (1b) ceil(x) = n if and only if n − 1 < x ≤ n (1c) floor(x) = n if and only if x − 1 < n ≤ x (1d) ceil(x) = n if and only if x ≤ n < x + 1 (2) x − 1 < floor(x) ≤ x ≤ ceil(x) < x + 1 (3a) floor(- x) = − ceil(x) (3b) ceil(- x) = − floor(x) (4a) floor(x + n) = floor(x) + n (4b) ceil(x + n) = ceil(x) + n

Definition of a Function

Let A and B be nonempty sets. A *function* f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f : A → B. Remarks: (1) Functions are sometimes also called mappings or transformations. (2) If f is a function from A to B, we say that f *maps* A to B. (3) A function f : A → B can also be defined in terms of a relation from A to B. Recall that a relation from A to B is just a subset of A × B. Hence, a relation from A to B that contains one, and only one, ordered pair (a, b) for every element a ∈ A, defines a function f from A to B. This function is defined by the assignment f(a) = b, where (a, b) is the unique ordered pair in the relation that has a as its first element.

Definition of the Identity Function

Let A be a set. The *identity function* on A is the function ι_A : A → A, where ι_A(x) = x for all x ∈ A. Remarks: (1) In other words, the identity function ι_A is the function that assigns each element to itself. (2) The function ιA is one-to-one and onto, so it is a bijection.

Definition of the Characteristic Function of a set S

Let S be a subset of a universal set U. The *characteristic function* f_S of S is the function from U to the set {0, 1} such that f_S(x) = 1 if x belongs to S and f_S(x) = 0 if x does not belong to S. Properties: (1) f_(A∩B)(x) = f_A(x) · f_B(x) (2) f_(A∪B)(x) = f_A(x) + f_B(x) − f_A(x) · f_B(x) (3) f_A^c(x) = 1 − f_A(x) (4) f_(A⊕B)(x) = f_A(x) + f_B(x) − 2f_A(x)f_B(x)

Definition of the Image of a Subset of A for a Function from A to B

Let f be a function from A to B and let S be a subset of A. The image of S under the function f is the subset of B that consists of the images of the elements of S. We denote the image of S by f(S), so f(S) = {t | ∃(s ∈ S)(t = f(s))} We also use the shorthand {f(s) | s ∈ S} to denote this set. Remarks: (1) Here, f(S) denotes a set, and *not* the value of the function f for the set S. (2) Some important properties: f(S ∪ T) = f(S) ∪ f(T) f(S ∩ T) ⊆ f(S) ∩ f(T) If f is one-to-one, then f(S ∩ T) = f(S) ∩ f(T)

Definition of the Inverse Image of a Function f

Let f be a function from the set A to the set B. Let S be a subset of B. We define the *inverse image* of S to be the subset of A whose elements are precisely all pre-images of all elements of S. We denote the inverse image of S by f^(−1)(S), so f^(−1)(S) = {a ∈ A | f(a) ∈ S} Remarks: (1) The notation f^(−1) is used in two different ways. Do not confuse the notation introduced here with the notation f^(−1)(y) for the value at y of the inverse of the invertible function f. (2) Notice also that f^(−1)(S), the inverse image of the set S, makes sense for all functions f, not just invertible functions. (3) Some important properties: f^(−1)(S ∪ T) = f^(−1)(S) ∪ f^(−1)(T) f^(−1)(S ∩ T) = f^(−1)(S) ∩ f^(−1)(T) f^(−1)(S^c) = (f^(−1)(S))^c

Definition of an Inverse Function

Let f be a one-to-one correspondence from the set A to the set B. The inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a) = b, hence f^(−1)(b) = a when f(a) = b (the inverse function of f is denoted by f ^(−1).

Addition and Multiplication of Real-Valued Functions

Let f_1 and f_2 be functions from A to R. Then, f_1 + f_2 and f_1 * f_2 are also functions from A to R defined for all x ∈ A by (f_1 + f_2)(x) = f_1(x) + f_2(x) (f_1 ⋅ f_2)(x) = f_1(x) ⋅ f_2(x) Note that the functions f_1 + f_2 and f_1 ⋅ f_2 have been defined by specifying their values at x in terms of the values of f_1 and f_2 at x.

Definition of the Composition of a Function

Let g be a function from the set A to the set B and let f be a function from the set B to the set C. The composition of the functions f and g, denoted for all a ∈ A by f ◦ g, is defined by (f ◦ g)(a) = f(g(a)) Remarks: (1) In other words, f ◦ g is the function that assigns to the element a of A the element assigned by f to g(a). That is, to find (f ◦ g)(a) we first apply the function g to a to obtain g(a) and then we apply the function f to the result g(a) to obtain (f ◦ g)(a) = f(g(a)). (2) The composition f ◦ g cannot be defined unless the range of g is a subset of the domain of f. Also, it is imperative that the domain of g be adhered to. (3) f ◦ g and g ◦ f are not necessarily equal; the commutative law does not hold for the composition of functions.

Definition of the Factorial Function

The *factorial function* is the function f : N → Z+, denoted by f(n) = n!. The value of f(n) = n! is the product of the first n positive integers, so f(n) = 1 · 2 ··· (n − 1) · n Remarks: (1) f(0) = 0! = 1 (2) Stirling's formula tells us that n! ∼ √(2πn)(n/e)^n Here, we have used the notation f (n) ∼ g(n), which means that the ratio f(n)/g(n) approaches 1 as n grows without bound, that is, lim (n → ∞) f(n)/g(n) = 1 The symbol ∼ is read "is asymptotic to."

Function Transformations for functions from R to R.

Suppose that f : R → R and suppose that a, b, c, and d ∈ R are constants. y = a · f(b(x + c)) + d (1) a is vertical stretch/compression |a| > 1 stretches |a| < 1 compresses a < 0 flips the graph upside down (2) b is horizontal stretch/compression |b| > 1 compresses |b| < 1 stretches b < 0 flips the graph left-right (3) c is horizontal shift c < 0 shifts to the right c > 0 shifts to the left (4) d is vertical shift d > 0 shifts upward d < 0 shifts downward

Definition of the Ceiling Function

The *ceiling function* assigns to the real number x the smallest integer that is greater than or equal to x. The value of the ceiling function at x is denoted by ceil(x). Remarks: (1) When considering statements about the ceiling function, it is useful to write x = n − ε, where n = ceil(x) is an integer and 0 ≤ ε < 1.

Definition of the Floor Function

The *floor function* assigns to the real number x the largest integer that is less than or equal to x. The value of the floor function at x is denoted by |_x_| or floor(x). Remarks: (1) The floor function is often also called the *greatest integer function*. It is often denoted by [x]. (2) A useful approach for considering statements about the floor function is to let x = n + ε,where n = floor(x) is an integer, and, the fractional part of x, satisfies the inequality 0 ≤ ε < 1.

Definition of the Range of a Function

The *range*, or *image*, of f is the set of all images of elements of A, that is, given f : A → B, the range of f, denoted f(A), is given by f(A) = {y ∈ B | ∃(x ∈ A)(y = f(x))} Remarks: (1) f(A) ⊆ B (2) f(A ∩ B) ⊆ f(A) ∩ f(B) (3) f(A ∪ B) = f(A) ∪ f(B)

Definition of a Bijective Function

The function f is a *one-to-one correspondence*, or a *bijection*, if it is both one to-one and onto. We also say that such a function is *bijective*. Remarks: (1) A one-to-one correspondence is called invertible because we can define an inverse of this function.

Definition of Equality of Functions

Two functions are equal when they (1) Have the same domain (2) Have the same codomain (3) Map each element of their common domain to the same element in their common codomain. Remarks: (1) If we change either the domain or the codomain of a function, then we obtain a different function (2) If we change the mapping of elements, then we also obtain a different function.

Definition of the Graph of a Function

We can associate a set of pairs in A × B to each function from A to B. This set of pairs is called the graph of the function and is often displayed pictorially to aid in understanding the behavior of the function. More formally, 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 ∈ A and f(a) = b}.

If both f and g are onto functions, then f ◦ g is also...

onto.