Section 2.1: Basics of Functions and Their Graphs

Obtaining Information from Graphs.

(Obtain information about a function from its graph.) You can obtain information about a function from its graph. At the right or left of a graph, you will find closed dots, open dots, or arrows. • A closed dot indicates that the graph does not extend beyond this point and the point belongs to the graph. • An open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph. • An arrow indicates that the graph extends indefinitely in the direction in which the arrow points.

Functions.

(Determine whether a relation is a function.) Table 2.1, based on our earlier discussion, shows the highest-paid TV celebrities and their earnings between June 2013 and June 2014, in millions of dollars. We've used this information to define two relations. Figure 2.3(a) shows a correspondence between celebrities and their earnings. Figure 2.3(b) shows a correspondence between earnings and celebrities. A relation in which each member of the domain corresponds to exactly one member of the range is a function. Can you see that the relation in Figure 2.3(a) is a function? Each celebrity in the domain corresponds to exactly one earning amount in the range. If we know the celebrity, we can be sure of his or her earnings. Notice that more than one element in the domain can correspond to the same element in the range: Stern and Cowell both earned $95 million. Winfrey and McGraw both earned $82 million. Is the relation in Figure 2.3(b) a function? Does each member of the domain correspond to precisely one member of the range? This relation is not a function because there are members of the domain that correspond to two different members of the range. The member of the domain 95 corresponds to both Stern and Cowell. The member of the domain 82 corresponds to both Winfrey and McGraw. If we know that the earnings are $95 million or $82 million, we cannot be sure of the celebrity. Because a function is a relation in which no two ordered pairs have the same first component and different second components, the ordered pairs (95, Stern) and (95, Cowell) are not ordered pairs of a function. Similarly, (82, Winfrey) and (82, McGraw) are not ordered pairs of a function. ​

Functions as Equations.

(Determine whether an equation represents a function.) Functions are usually given in terms of equations rather than as sets of ordered pairs. For example, here is an equation that models the percentage of first-year college women claiming no religious affiliation as a function of time: y = 0.012x² - 0.2x + 8.7 The variable x represents the number of years after 1970. The variable y represents the percentage of first-year college women claiming no religious affiliation. The variable y is a function of the variable x. For each value of x, there is one and only one value of y. The variable x is called the independent variable because it can be assigned any value from the domain. Thus, x can be assigned any nonnegative integer representing the number of years after 1970. The variable y is called the dependent variable because its value depends on x. The percentage claiming no religious affiliation depends on the number of years after 1970. The value of the dependent variable, y, is calculated after selecting a value for the independent variable, x. We have seen that not every set of ordered pairs defines a function. Similarly, not all equations with the variables x and y define functions. If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a function of x. ​

Function Notation.

(Evaluate a function) If an equation in x and y gives one and only one value of y for each value of x, then the variable y is a function of the variable x. When an equation represents a function, the function is often named by a letter such as f, g, h, F, G, or H. Any letter can be used to name a function. Suppose that f names a function. Think of the domain as the set of the function's inputs and the range as the set of the function's outputs. As shown in Figure 2.5, input is represented by x and the output by f(x). The special notation f(x), read "f of x" or "f at x," represents the value of the function at the number x. Let's make this clearer by considering a specific example. We know that the equation y = 0.012x² - 0.2x + 8.7 defines y as a function of x. We'll name the function f. Now, we can apply our new function notation. ​

Relations.

(Find the domain and range of a relation.) Forbes magazine published a list of the highest-paid TV celebrities between June 2013 and June 2014. The results are shown in Figure 2.1. The graph indicates a correspondence between a TV celebrity and that person's earnings, in millions of dollars. We can write this correspondence using a set of ordered pairs: {(Stern, 95), (Cowell, 95), (Beck, 90), (Winfrey, 82), (McGraw, 82)} These braces ( ) indicate we are representing a set. The mathematical term for a set of ordered pairs is a relation.

Graphs of Functions.

(Graph functions by plotting points.) The graph of a function is the graph of its ordered pairs. For example, the graph of f(x) = 2x is the set of points (x,y) in the rectangular coordinate system satisfying y = 2x. Similarly, the graph of g(x) = 2x + 4 is the set of points (x,y) in the rectangular coordinate system satisfying the equation y = 2x + 4. In the next example, we graph both of these functions in the same rectangular coordinate system.

Identifying Intercepts from a Function's Graph.

(Identify intercepts from a function's graph.) Figure 2.15 illustrates how we can identify intercepts from a function's graph. To find the x-intercepts, look for the points at which the graph crosses the x-axis. There are three such points: (-2,0), (3, 0), and (5, 0). Thus, the x-intercepts are -2, 3, 5. We express this in function notation by writing f(-2) = 0, f(3) = 0, f(5) = 0. We say that -2, 3, and 5 are the zeros of the function. The zeros of a function f are the x-values for which f(x) = 0. Thus, the real zeros are the x-intercepts. To find the y-intercept, look for the point at which the graph crosses the y-axis. This occurs at (0, 3). Thus, the y-intercept is 3. We express this in function notation by writing f(0) = 3. By the definition of a function, for each value of x we can have at most one value for y. What does this mean in terms of intercepts? A function can have more than one x-intercept but at most one y-intercept. ​

Identifying Domain and Range from a Function's Graph.

(Identify the domain and range of a function from its graph.) Figure 2.13 illustrates how the graph of a function is used to determine the function's domain and its range. Domain: set of inputs (found on x-axis) Range: set of outputs (found on y-axis) Let's apply these ideas to the graph of the function shown in Figure 2.14. To find the domain, look for all the inputs on the x-axis that correspond to points on the graph. Can you see that they extend from -4 to 2, inclusive? The function's domain can be represented as follows: USING SET-BUILDER NOTATION {x | -4 <= x <= 2} USING INTERVAL NOTATION [-4, 2] To find the range, look for all the outputs on the y-axis that correspond to points on the graph. They extend from 1 to 4, inclusive. The function's range can be represented as follows: SET-BUILDER NOTATION {y | 1 <= y <= 4} The square brackets inducate the numbers are included. ​

The Vertical Line Test.

(Use the vertical line test to identify functions.) Not every graph in the rectangular coordinate system is the graph of a function. The definition of a function specifies that no value of x can be paired with two or more different values of y. Consequently, if a graph contains two or more different points with the same first coordinate, the graph cannot represent a function. This is illustrated in Figure 2.8. Observe that points sharing a common first coordinate are vertically above or below each other. FIGURE 2.8 y is not a function of x because 0 is paired with three values of y, namely, 1, 0, and -1. This observation is the basis of a useful test for determining whether a graph defines y as a function of x. The test is called the vertical line test. ​

Definition of a Function.

A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range.

Definition of a Relation.

A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation. Find the domain and range of the relation: {(Stern, 95), (Cowell, 95), (Beck, 90), (Winfrey, 82), (McGraw, 82)} Solution: The domain is the set of all first components. Thus, the domain is: {Stern, Cowell, Beck, Winfrey, McGraw} The Range is the set of all second components. Thus, the range is: {95, 90, 82}. It is not necessary to list multiples of the same number.

Example 5 Graphing Functions.

Graph the function f(x) = 2x and g(x) = 2x + 4 in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. SOLUTION We begin by setting up a partial table of coordinates for each function. Then we plot the five points in each table and connect them, as shown in Figure 2.7. The graph of each function is a straight line. Do you see a relationship between the two graphs? The graph of g is the graph of f shifted vertically up by 4 units. The graphs in Example 5 are straight lines. All functions with equations of the form f(x) = mx + b graph as straight lines. Such functions, called linear functions, will be discussed in detail in Section 2.3.

If I reverse a function's components, will this new relation be a function?

If a relation is a function, reversing the components in each of its ordered pairs may result in a relation that is not a function.

The Vertical Line Test for Functions.

If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

The range in Example 8(e) was identified as {y | y = 1,2,3} Why didn't you also use interval notation like you did in the other parts of Example 8?

Interval notation is not appropriate for describing a set of distinct numbers such as {1, 2, 3}. Interval notation, [1, 3], would mean that numbers such as 1.5 and 2.99 are in the range, but they are not. That's why we only used set-builder notation.

Example 7 Analyzing the Graph of a Function.

The human immunodeficiency virus, or HIV, infects and kills helper T cells. Because T cells stimulate the immune system to produce antibodies, their destruction disables the body's defenses against other pathogens. By counting the number of T cells that remain active in the body, the progression of HIV can be monitored. The fewer helper T cells, the more advanced the disease. Figure 2.9 shows a graph that is used to monitor the average progression of the disease. The average number of T cells, f(x), is a function of time after infection, x. a. Explain why f represents the graph of a function. b. Use the graph to find and interpret f(8). c. For what value of x is f(x) = 350? d. Describe the general trend shown by the graph. SOLUTION a. No vertical line can be drawn that intersects the graph of f more than once. By the vertical line test, f represents the graph of a function. b. To find f(8) or f of 8, we locate 8 on the x-axis. Figure 2.10 shows the point on the graph of f for which 8 is the first coordinate. From this point, we look to the y-axis to find the corresponding y-coordinate. We see that the y-coordinate is 200. Thus, f(8) = 200. When the time after infection is 8 years, the average T cell count is 200 cells per milliliter of blood. (AIDS clinical diagnosis is given at a T cell count of 200 or below.) c. To find the value of x for which f(x) = 350, we find the approximate location of 350 on the y-axis. Figure 2.11 shows that there is one point on the graph of f for which 350 is the second coordinate. From this point, we look to the x-axis to find the corresponding x-coordinate. We see that the x-coordinate is 6. Thus, f(x) = 350 for x = 6.

Doesn't f(x) indicate that I need to multiply f and x?

The notation f(x) does not mean "f times x." The notation describes the value of the function at x.