Calculus

Chapter 1.1 - Functions Defined

Definition:

A set is a collection of objects. In mathematics, these objects are usually numbers. We denote the beginning of a set by the symbol "{" and the end of a set by the symbol "}". Each object in the set is called an element of the set. Each element of the set is separated by a comma.

Example:

The set {1,4,6,2,7,9} represents a collection of 6 numbers. In a set, the order is not a concern. Two sets are said to be equal if they have the same elements. The above set is called a finite set. Sets can be infinite. For example, consider the set of the Natural Numbers {1,2,3,4,5,...}. Recall the notation "..." represents that the obvious pattern repeats indefinitely. Sets are convenient notation, but not all sets of numbers can be listed. For example, suppose we wanted to describe the set of all real numbers between 0 and 1 inclusively. Using the previous set notation will not suffice because there is no pattern that can be found to list all those numbers (this was proved more generally by a mathematician named Cantor). This is why we have what is called interval notation. The notation (0,1) represents all real numbers between 0 and 1 exclusively. If we wanted to include 0 and 1 in the set, we need to use square brackets instead of the parenthesis. Note: (0,1] represents all real numbers between 0 and 1 including 1 but not 0. We can also modify our set notation to describe intervals. We could write {x | 0 < x < 1}, which is read as "the set of all numbers x such that x is greater than or equal to 0 and less than one", to describe the set of all numbers between 0 and 1 including 0 but not 1. Note: the bar "|" in words is read as "such that".

Definition:

A function is a rule that assigns a value to each element in a set A exactly one element in a set B. Set A is what we call the domain of the function, and set B is what we call the range of the function. The following diagram illustrates this idea.

Example:

Look at the two mappings below. The mapping on the left is a function because it follows the definition - each element in set A gets mapped to exactly one element in set B. The map on the right is not a function because element b in set A gets mapped to more than one element in set B, namely x and y.

Domain, Co-domain, and Image:

Set A is what we call the domain of a function - the set of all values of x that get mapped to some number by the function. We denote the domain of a function (x) by Domain(). When dealing with functions, if the domain is not specified, then assume the domain of the function to be where ever it is defined over the real numbers. For example, consider the function (x) = x + 1. Since a domain is not specified, we assume the domain to be all real numbers (we know how to add 1 to any real number). Sometimes a domain is specified. For example, consider the function (x) = x + 1 with domain [0, ) (Note: recall this is interval notation meaning any real number between 0 and infinity including 0). Even though this function has the same equation as the previous function, they are different functions because their domains are different.

The co-domain of a function is the set of all possible numbers an element in the domain can get mapped to. The image of a number in the domain of a function is what the function maps that number to in the range, specifically (x).

Example:

Let (x) = x². Since is defined for all real values (we can square any real number), we will assume our domain to be the real numbers. If x = 2, then the image of x = 2 under the function is simply (2) = 2² = 4. So the image is 4.

Question:

      Find the image of x = -1 under the function (x) = 2x+1.

-1
  2
-2
  1

Range:

The range of a function is the set of all numbers that every possible value in the domain gets mapped to. For example, let (x) = x². The domain is assumed to be all real numbers (we can square any real number). Since 2 is in the domain and (2) = 4, then 4 is in the range of since 2 got mapped to 4 under the function (x). In set notation, the range of (x) denoted by Range() is given by Range() = {(x) | x Domain((x))}. We read this notation as: The range of (x) is equal to the set of all elements (x) such that x is an element in the domain D of the function . This set is just the collection of images of every possible value x in the domain under the function . Some books treat the co-domain and the range of a function synonymously, but their is a technical distinction. The co-domain is the set of all possible elements each element in the domain can get mapped to, where as the range is the only numbers each element in the domain can get mapped to. Technically, the range is a subset of the co-domain.

Example:

Consider the function (x) = x. Let's find some images of x under . If x = -100 we have that (x) = -100. If x = 0 we have that (0) = 0. If x = 100 we have that (100) = 100. It seems that for every value we plug in, we get back. Since our domain is , then the set of all images of x in the domain of are going to contain every element of - the symbol denoting the real numbers. So our range must be . We denote this one of two ways: Range() = , or Range() = (-, ).

Question:

Let (x) = x². Find the range of .

[-, ]
(0, )
(-, 0)
[0, )

Determining the Domain and Range by graphing:

The domain and range of a function will not always be . For example, one could define a function (x) = 2x with Domain() = [0,1] - that is all real numbers between 0 and 1 including 0 and 1. Verify that the range of the given function is [0,2]. That is, for every number in the range [0,2], there is some number in [0,1] that gets mapped to it under the function (x)=2x. Not all functions are defined for every real number. For example, (x) = 1/x is defined for every real number except 0. So the domain of is the set of all real numbers except 0, in notation (-, 0)(0, )

Example:

Consider the function (x) = . We see that (x) is only defined for non-negative values of x. So our domain is [0, ). If we find images of x in [0, ), we see that the Range() = [0, ). Let's graph this function and see how the domain and range appear graphically.

The graph suggests that the domain is [0,) because the x-coordinates of the graph are only present for those values. The graph also suggests that the range is [0,) because the y-coordinates of the graph are only present for those values.

Question:

      Let (x) = .  Find the domain and range of .

Domain=(- , )  Range=(- ,)
Domain=(0, )  Range=(0, )
Domain=(- , 0] [0, )   Range=(-, 0] [0, )
Domain=(- , 0) (0, )  Range=(-, 0) (0, )

Vertical Line Test:

If you look back at the definition of a function, you will see that a function cannot map a value x = a from the domain to two values in the co-domain. Thus, graphically we can test a graph to see if it is a function by what is called the vertical line test. We do this by actually running a vertical line through all parts of the graph and see if it intersects the graph more than once. Look at the examples below.

Notice that the graph on the left is a function because the graph will not be intersected more than once given a vertical line. The graph on the right however is not a function because there are vertical lines that intersect the graph at several points.

X and Y Intercepts:

A x-intercept of a function is the coordinate(s) point in which the graph of the function crosses the x-axis. A y-intercept of a function is the coordinate(s) point in which the graph of the function crosses the y-axis. One can solve for the x-coordinate of the x-intercept by letting y = 0. One can solve for the y-coordinate of the y-intercept by letting x = 0.

Example:

Consider the function (x) = x² - 1. Let y = x² - 1. If we let x = 0, we have that y = -1. So the y-intercept is given by the coordinate point (0,-1). Now let y = 0. Then we have that 0 = x² - 1. Solving for x, we have that x = 1. So the x-intercepts are given by (-1,0) and (1,0). Look at the graph below to confirm our results.

Question:

Determine the x and y intercepts of the function g(x) = 1 - .

x-int: (-1,0) and y-int: (0,-1)
x-int: (-1,0) and y-int: (0,1)
x-int: (1,0) and y-int: (0,-1)
x-int: (1,0) and y-int: (0,1)