University Mathematics Handbook (2015)

II. Functions

Chapter 1. General Properties of Functions

1.1  Definition

Let and be two sets of real numbers.

A function is a given rule fitting each number in set , a one and only one number of set .

That function is denoted as , which says of , or , which reads is into .

Set is called the function's domain.

is called an input variable or argument.

If , then is said to be the image of , or is the preimage of .

If is described by an algebraic expression, then the maximum set of real numbers for which the expression exits is the natural domain of .

is bounded on domain if there exists a number such that for all .

Example: The domain of is , .

The graph of function is a set of all pairs of coordinates where belongs to .

1.2  Increasing and Decreasing Functions

The function is increasing in domain if for every two points and from , and where there holds .

The function is decreasing in domain if for every two points and , which are members of , and where there holds .

If the inequality is strict, then the function is strictly increasing (or decreasing).

Example: The function is increasing in the domain and decreasing in , and is neither increasing nor decreasing in any interval containing the point .

The graph of an increasing function in an interval is a curve ascending from left to right.

The graph of a decreasing function in an interval is a curve descending from left to right.

1.3  Odd and Even Functions

is called an even function if for all in its domain there is also in its domain, such that .

Example: , , and are even functions.

is called an odd function, if for all in its domain there is also in its domain, such that .

Example: , , and are odd functions.

Example: is neither an even nor an odd function.

The sum of two even (or odd) functions is an even (or odd) function.

The product of two even (or odd) functions is an even function.

The product of an even function and an odd function is an odd function.

A graph of an even function is symmetric about the axis.

A graph of an odd function is symmetric about the origin.

1.4  Periodic Function

is called a periodic function if there exists a positive number such that for all there is also and there holds .

The minimum , if it exists, is called the period of .

Example: , and are periodic functions with a period.

1.5  One-to-One Correspondence Function (Bijective Function)

is a one-to-one correspondence function if for all of there exists at most one of such that .

In other words, is a one-to-one correspondence function if for all of if then .

Example: is a bijective function in the domain and not bijective in , since for , there holds .

1.6  Surjective Function

The function is a surjective function if for all there exists a such that .

Examples:

a.   is a surjective but not one-to-one correspondence function.

b.   is both a surjective and a one-to-one correspondence function.

1.7  Inverse Function

Let . If for every of there exists a unique of such that , then is called an invertible function, and is the inverse function of . That is, and .

Examples:

a.  The inverse function of is

b.  The inverse function of , is

In the inverse function the input variable is and the output variable is . It's more convenient to have here as the input variable, so and are switched, and the result is the inverse function .

In the previous example, the inverse function of was .

Theorem: If is invertible and is its inverse function, then the graphs of these two functions are symmetric to each other about the line
(Figure 8).

Theorem: The function is invertible if, and only if, it is a one-to-one correspondence and a surjective function.

1.8  Equivalent Sets

a.  Set is called equivalent to set if there exists a one-to-one correspondence and surjective function .

Example: Interval is equivalent to as one-to-one correspondence function mapping on .

b.  A set equivalent to natural numbers set is called a countable set.

Example: Sets and are countable sets.

c.  A set equivalent to interval is called a linear continuum.

Example: Open or closed intervals, , and are continuum.

1.9  Operations with Functions

Let , .

Functions , and are only defined when belongs to as well as to , and therefore, are defined in domain .

To compose and , when , is function , which results from .

Order of operations is significant. First, is applied to and, only then, is applied to .

Example: , , therefore

,

1.10  Elementary Functions

The functions, resulting from the functions (constant), , , by their addition, subtraction, multiplication, and partition, their compositions and their inverses, are elementary functions.

Examples:

a.  Function is elementary since .

b.   and therefore, is an elementary function.