Learning objectives
 Determine if a relation is a function or not
 Determine the domain and range of a function
 Graph, or read a graph of basic real functions on subintervals of their domains
 Determine the average rate of change of a function
Introduction
Consider two situations shown in the boxes below:
Situation 1: You are selling candy bars for a school fundraiser. Each candy bar costs $3.00  Situation 2: You collect data from several students in your class on their ages and their heights: (18,65"), (17,64"), (18,67"), (18,68"), (17,66") 

In the first situation, let the variable x represent the number of candy bars that you sell, and let y represent the amount of money you make. If you sell x candy bars, you will make dollars. For example, if you sell 25 candy bars, you will make 3(25) = $75.00. Notice that you can use the number of candy bars you sell to “predict” how much money you will make.
Now consider the second situation. Can you use the data to predict height, given age?
This is not the case in the second situation. For example, if a student is 18 years old, there are several heights that the student could be.
The first situation is an example of a function, and the second example is not a function. In this lesson you will learn to identify mathematical relations that are functions and to describe them in particular ways. This lesson will prepare you to learn about several kinds of functions throughout this chapter.
Relations and Functions
Both situations above are relations. A relation is simply a relationship between two sets of numbers or data. For example, in the second situation, we created a relationship between students’ ages and heights, just by writing each student’s information as an ordered pair. In the first situation, there is a relationship between the number of candy bars you sell and the amount of money you make. The first example is different from the second because it represents a function. A more specific type of relation is a function. A function is a relation where there is exactly one output for every input. Simply stated, the input values of a function cannot repeat.
We can represent functions in many ways. Some of the most common ways to represent functions include: sets of ordered pairs, equations, and graphs. The figure below shows the same function depicted in each of these representations:
Representation  Example 

Set of ordered pairs  (1,3), (2,6), (3,9), (4,12) (a subset of the ordered pairs for this function) 
Equation  
Graph 
 In contrast, the relation shown in the figure below is not a function:
Representation  Example 

Set of ordered pairs  (4, 2), (4, 2), (9, 3), (9, 3) (A subset of the ordered pairs for this relation) 
Equation  
Graph 
To verify that this relation is not a function, we must show that at least one x value is paired with more than one y value. If you look at the first representation, the set of ordered pairs, you can see that 4 is paired with 2 and with 2. Similarly, 9 is paired with 3 and with 3. Therefore the relation is not a function. It is harder to identify a function from an equation, unless you are familiar with different kinds of equations. (If you are not already familiar with different kinds of functions and their graphs, you will be by the end of this chapter!) If we look at the graph above, we can see that, except for x = 0, the x values of the relation are each paired with two y values. Therefore the relation is not a function.
One way to determine whether a relation is a function when looking at a graph is by doing a "vertical line test". If a vertical line can be drawn anywhere on the graph such that the line crosses the relation in two places, then the relation is not a function. If all possible vertical lines will only cross the relation in one place, then the relation is a function. This works because if a vertical line crosses a relation in more than one place it means that there must be two y values corresponding to one x value in that relation. For example, the graph below of y=3x shows it is a function because any vertical line that is drawn only crosses the relation in one place.
Conversely, the graph below of x=y^{2} shows it is not a function because a vertical line can be drawn that crosses the relation in two places.
Example 1: Determine if the relation is a function or not
a. (2, 4), (3, 9), (5, 11), (5, 12)  b. 

Solution:
a. (2, 4), (3, 9), (5, 11), (5, 12)
 This relation is not a function because 5 is paired with 11 and with 12.
b. This relation is a function because every x is paired with only one y.
Once you are able to determine if a relation is a function, you should then be able to state the set of x values and the set of y values of the function. Next we will define these sets as the domain and the range of a function, and we will look at some specific examples of functions and their domains and ranges.
Independent variable, domain
The domain of a function is defined as the set of all x values for which the function is defined. For example, the domain of the function is the set of all real numbers, often written as . This means that x can be any real number. Other functions have restricted domains. For example, the domain of the function is the set of all real numbers greater than or equal to zero. The domain of this function is restricted in this way because the square root of a negative number is not a real number. Therefore, the domain is restricted to nonnegative values of x so that the function values will be defined.
It is often easy to determine the domain of a function by (1) considering what restrictions there might be and (2) looking at a graph. For example, we can see that the function , shown in the graph below, has a domain of all real numbers greater than or equal to zero because the graph only exists for x values that are greater than or equal to zero.
Example 2: State the domain of each function:
a.  b.  c. (2, 4), (3, 9), (5, 11) 

Solution:
a.
The domain of this function is the set of all real numbers. There are no restrictions.
b.
The domain of this function is the set of all real numbers except x = 0. The domain is restricted this way because a fraction with denominator zero is undefined.
c. (2, 4), (3, 9), (5, 11)
The domain of this function is the set of x values: {2, 3, 5}.
The variable x is often referred to as the independent variable, while the variable y is referred to as the dependent variable. We talk about x and y this way because the y values of a function depend on what the x values are. That is why we also say that “y is a function of x.” For example, the value of y in the function y=3x depends on what x value we are considering. If x = 4, we can easily determine that y = 3(4) = 12. Returning to the situation in the introduction, we can say that the amount of money you take in depends on the number of candy bars you sell.
When we are working with a function in the form of an equation, there is a special notation we can use to emphasize the fact that y is a function of x. For example, the equation can also be written as . It is important to remember that represents the y values, or the function values, and that the letter f is not a variable. (That is, f(x) does not mean that we are multiplying.)
Now that we have considered the domain of a function, we will turn to the range.
Dependent Variable, Range
The range of a function is defined as the set of all y values for which a function is defined. Just as we did with domain, we can examine a function and determine its range. Again, it is often helpful to think about what restrictions there might be, and what the graph of the function looks like. Consider for example the function . The domain of this function is , all real numbers, but what about the range?
The range of the function is the set of all real numbers greater than or equal to zero. This is the case because every y value is the square of an x value. If we square positive and negative numbers, the result will always be positive. If x = 0, then y = 0. We can also see the range if we look at a graph of . Notice below that the y values are all greater than or equal to zero.
Example 3: State the domain and range of the function
Solution: The domain and range of the function
For this function, we can choose any x value except x = 0 . Therefore the domain of the function is the set of all real numbers except x = 0.
The range is also restricted to the nonzero real numbers, but for a different reason. Because the numerator of the fraction is 2, the numerator can never equal zero, so the fraction can never equal zero.
Now that we have defined what it means for a relation to be a function, and we have defined domain and range of a function, we can look at some specific examples of functions and their graphs.
Real values, intervals
A function is defined as a real function if both the domain and the range are sets of real numbers. Many of the functions you have likely encountered before are real functions, and many of these functions have Domain = . Consider, for example, the function . The graph of this function, shown at the beginning of this lesson, is shown again below, slightly varied.
You may already be familiar with the graphs of lines. In particular, you may already be in the habit of placing arrows at the ends. We do this in order to indicate that the line will continue forever in both the positive and negative directions, both in terms of the domain and the range. The line above, however, only shows the function on the interval [3, 3]. The square brackets indicate that the graph includes the endpoints of the interval x = 3 and x = 3. We call this a closed interval. A closed interval contains its endpoints. In contrast, an open interval does not contain its endpoints. We indicate an open interval with parentheses. For example, (3, 3) indicates the set of numbers between 3 and 3, not including 3 and 3. You may have noticed that the open interval notation looks like the notation for a point (x, y) in the plane. It is important to read an example or a homework problem carefully to avoid confusing a point with an interval!
When you are graphing functions in general, especially those for which the domain is the set of all real numbers (D = ), you will necessarily be showing a subset of the domain. In studying functions, you may also be asked to describe what the graph of a function looks like on a particular subset of its domain. The table below summarizes the kinds of intervals you may need to consider while studying functions and their domains:
Interval notation  Inequality notation  Description 

The value of x is between a and b, including a and b, where a, b are real numbers.  
The value of x is between a and b, not including a and b.  
The value of x is between a and b, including a, but not including b.  
The value of x is between a and b, including b, but not including a.  
The value of x is strictly greater than a.  
The value of x is greater than or equal to a  
The value of x is strictly less than a  
The value of x is less than or equal to a. 
Example 4: Sketch the graph of the function on the interval [4, 12)
Solution: The figure below shows a graph of on the given interval:
Above we have considered several different kinds of functions. If you have looked at the graphs of functions such as and , you may have noticed that the shapes of the graphs are quite different. Later in the chapter you will study different “families” of functions and their graphs. Here we will consider one aspect of functions that requires us to consider specific intervals of the function.
Average rate of change
Consider the following situation: you are on a week long road trip with your friend. When you begin to drive on the second day, you have already driven a total of 200 miles. After 6 hours of driving on the second day, you have driven a total of 500 miles. On average, how many miles did you drive per hour on the second day of the trip?
The graph below shows this situation, with the x axis representing the number of hours driving (on the second day), and the y axis representing the number of miles driven. The first point on the graph, (0,200), says that at the beginning of the second day you have already driven 200 miles. The second point on the graph, (6,500), says that after 6 hours of driving on the second day you have driven 500 miles total.
Notice that in total, during your 6 hours of driving, you driven 300 miles. The rate at which you drove is 300 miles in 6 hours, or 50 miles per hour. We refer to this rate as the average rate of change because it is an average across the 6 hours. That is, you did not necessarily drive 50 miles every hour. There could have been one hour where you drove 70 miles and another hour where you drove only 30 miles.
We can represent the average rate of change on the graph by indicating how much each quantity has changed: The y values increased by 300, and the x values increased by 6. The average rate of change is the ratio of these changes in each variable. This is how we can define average rate of change in general:
We can examine the average rate of change of a function, whether it is represented as data, as in the previous example, or by an equation.
Example 5: Find the average rate of change of each function on the given interval
a. on [0, 2]  b. on [1, 7] 

Solution:
a.
The endpoints of the interval are (0,0) and (2,4). Therefore the change in y is 4 and the change in x is 2. The average rate of change is 4/2 = 2.
b.
The endpoints of the interval are (1,4) and (7,28). Therefore the change in y is 28  4 = 24 and the change in x is 7  1 = 6. The average rate of change is 24/6 = 4.
Notice that the average rate of change of the function is the slope of the line, 4. While a linear function has a constant slope, other functions, such as , will not. You will explore this idea in greater detail in your study of calculus.
Lesson Summary
In this lesson we have defined what it means for a relation between two variables (such as x and y) to be a function, and we have considered several key aspects of functions: the domain and range of a function, subintervals of the domain of a real function, and the average rate of change of a function. We have explored these ideas in the context of specific examples of linear and nonlinear functions. In the coming lessons you will rely on the ideas introduced here to further develop your understanding of functions.
Points to Consider
1. How can you determine if a relation is a function?
2. What kinds of functions have domain and range both equal to the set of all real numbers?
3. How is average rate of change different for linear and nonlinear functions?
Review Questions
 Determine if each relation is a function:

a. (1,4), (0, 3), (1, 5), (1, 7), (2, 15) b.  State the domain and range of each relation in question 1.
 Give an example of a function for which the domain and range are equivalent to each other.
 Write each interval using interval notation.

a. x>5 b. 4 ≤ x <7  Sketch a graph of the function on the interval [10, 10]
 What is the average rate of change of the function on the interval [2, 5]?
 Consider the function . a. Choose two points on the graph of and calculate the average rate of change. b. Choose two other points and calculate the average rate of change. c. Compare your answers in parts a and b.
 Consider the function a. If a > 0, what is the range of this function? b. If a < 0, what is the range of this function?
 Give an example of a relation that is not a function, and explain why it is not a function.
 Consider the function from question 8. Assume that a > 0. a. What is the average rate of change of the function on the interval [0, 1]? b. What is the average rate of change of the function on the interval [0, 0.1]?
Review Answers
 a. Not a function b. Is a function
 a. D: {1, 0, 1, 2}; R: {3, 4, 5, 7, 15} b. D:; R:
 Answers will vary. The function in 1b is an example.
 a. b. [4,7)
 The two points are (2, 12) and (5, 75). The average rate of change is 63/3 = 21.
 a. and b. Choice of points will vary, but the average rate of change for both a and b should be 2. c. They should be equal to 2, the slope of the function, because this is a line.
 a. If a > 0, R: y ≥ 0 b. If a < 0, R: y ≤ 0
 Answers will vary. Example: (2, 3) (2, 4) (4, 5) is not a function because the element 2 in the domain is paired with more than one element of the range.
 a. b.