3Functions from the Numerical, Algebraic, and Graphical Viewpoints The following table gives the approximate number of Facebook users at various times since its establishment early in 2004.Let’s write n(0) for the number of members (in millions) at time t = 0, n(1) for the number at time t = 1, and so on (We read n(0) as “n of 0”). Thus, n(0) = 0, n(1) = 1, n(2) = 5.5, , n(6) = 450.
4Functions from the Numerical, Algebraic, and Graphical Viewpoints In general, we write n(t) for the number of members (in millions) at time t. We call n a function of the variable t, meaning that for each value of t between 0 and 6, n gives us a single corresponding number n(t) (the number of members at that time).In general, we think of a function as a way of producing new objects from old ones. The functions we deal with in this text produce new numbers from old numbers.The numbers we have in mind are the real numbers, including not only positive and negative integers and fractions but also numbers like or .
5Functions from the Numerical, Algebraic, and Graphical Viewpoints For this reason, the functions we use are called real-valued functions of a real variable.For example, the function n takes the year since the start of 2004 as input and returns the number of Facebook members as output (Figure 1).The variable t is called the independent variable, while n is called the dependent variable as its value depends on t.Figure 1
6Functions from the Numerical, Algebraic, and Graphical Viewpoints A function may be specified in several different ways. Here, we have specified the function n numerically by giving the values of the function for a number of values of the independent variable, as in the table.The set of values of the independent variable for which a function is defined is called its domain and is a necessary part of the definition of the function.
7Functions from the Numerical, Algebraic, and Graphical Viewpoints Notice that the preceding table gives the value of n(t) at only some of the infinitely many possible values in the domain [0, 6].The domain of a function is not always specified explicitly; if no domain is specified for the function f, we take the domain to be the largest set of numbers x for which f (x) makes sense.This “largest possible domain” is sometimes called the natural domain.
8Functions from the Numerical, Algebraic, and Graphical Viewpoints The previous Facebook data can also be represented on a graph by plotting the given pairs of numbers (t, n(t)) in the xy-plane. (See Figure 2. We have connected successive points by line segments.)In general, the graph of a function f consists of all points (x, f (x)) in the plane with x in the domain of f.Figure 2
9Functions from the Numerical, Algebraic, and Graphical Viewpoints In Figure 2 we specified the function n graphically by using a graph to display its values.Suppose now that we had only the graph without the table of data. We could use the graph to find approximate values of n. For instance, to find n(5) from the graph, we do the following:1. Find the desired value of t at the bottom of the graph (t = 5 in this case).2. Estimate the height (n-coordinate) of the corresponding point on the graph (around 150 in this case).Thus, n(5) ≈ 150 million members.
10Functions from the Numerical, Algebraic, and Graphical Viewpoints In some cases we may be able to use an algebraic formula to calculate the function, and we say that the function is specified algebraically.These are not the only ways in which a function can be specified; for instance, it could also be specified verbally, as in “Let n(t) be the number of Facebook members, in millions, t years since the start of 2004.”Notice that any function can be represented graphically by plotting the points (x, f (x)) for a number of values of x in its domain.
11Functions from the Numerical, Algebraic, and Graphical Viewpoints A real-valued function f of a real-valued variable x assigns to each real number x in a specified set of numbers, called the domain of f, a unique real number f (x), read “f of x.”The variable x is called the independent variable, and f is called the dependent variable.A function is usually specified numerically using a table of values, graphically using a graph, or algebraically using a formula. The graph of a function consists of all points (x, f (x)) in the plane with x in the domain of f.
12Functions from the Numerical, Algebraic, and Graphical Viewpoints Quick ExampleA function specified numerically: Take c(t) to be the world emission of carbon dioxide in year t since 2000, represented by the following table:
13Functions from the Numerical, Algebraic, and Graphical Viewpoints The domain of c is [0, 30], the independent variable is t, the number of years since 2000, and the dependent variable is c, the world production of carbon dioxide in a given year. Some values of c are:c (0) = 24c(10) = 31c (30) = 4124 billion metric tons of CO2 were produced in 2000.31 billion metric tons of CO2 were produced in 2010.41 billion metric tons of CO2 were projected to be produced in 2030.
14Functions from the Numerical, Algebraic, and Graphical Viewpoints Graph of c: Plotting the pairs (t, c (t)) gives the following graph:
15Example 1 – iPod SalesThe total number of iPods sold by Apple up to the end of year x can be approximated byf (x) = 4x2 + 16x + 2 million iPods (0 ≤ x ≤ 6),where x = 0 represents 2003.a. What is the domain of f ? Compute f (0), f (2), f (4), and f (6). What do these answers tell you about iPod sales? Is f (–1) defined?b. Compute f (a), f (–b), f (a + h), and f (a) + h assuming that the quantities a, –b, and a + h are in the domain of f.c. Sketch the graph of f. Does the shape of the curve suggest that iPod sales were accelerating or decelerating?
16Example 1(a) – SolutionThe domain of f is the set of numbers x with 0 ≤ x ≤ 6—that is, the interval [0, 6].If we substitute 0 for x in the formula for f (x), we getf (0) = 4(0)2 + 16(0) + 2 = 2Similarly,f (2) = 4(2)2 + 16(2) + 2 = 50By the end of 2003 approximately 2 million iPods had been sold.By the end of 2005 approximately 50 million iPods had been sold.
17Example 1(a) – Solution f (4) = 4(4)2 + 16(4) + 2 = 130 cont’df (4) = 4(4)2 + 16(4) + 2 = 130f (6) = 4(6)2 + 16(6) + 2 = 242As –1 is not in the domain of f, f (–1) is not defined.By the end of 2007 approximately 130 million iPods had been sold.By the end of 2009 approximately 242 million iPods had been sold.
18Example 1(b) – Solutioncont’dTo find f (a) we substitute a for x in the formula for f (x) to getf (a) = 4a2 + 16a + 2.Similarly,f (–b) = 4(–b)2 + 16(–b) + 2= 4b2 – 16b + 2f (a + h) = 4(a + h)2 + 16(a + h) + 2= 4(a2 + 2ah + h2) + 16a + 16h + 2= 4a2 + 8ah + 4h2 + 16a + 16h + 2Substitute a for x.Substitute −b for x.(−b)2 = b2Substitute (a + h) for x.Expand.
19Example 1(b) – Solutioncont’df (a) + h = 4a2 + 16a h.Note how we placed parentheses around the quantities at which we evaluated the function.If we tried to do without any of these parentheses we would likely get an errors:Correct expression:f (a + h) = 4(a + h)2 + 16(a + h) + 2.Add h to f (a).NOT 4a + h2 + 16a + h + 2x
20Example 1(b) – Solutioncont’dAlso notice the distinction between f (a + h) and f (a) + h: To find f (a + h), we replace x by the quantity (a + h); to find f (a) + h we add h to f (a).
21Example 1(c) – Solutioncont’dTo draw the graph of f we plot points of the form (x, f (x)) for several values of x in the domain of f.Let us use the values we computed in part (a):
22Example 1(c) – Solutioncont’dGraphing these points gives the graph shown in Figure 3, suggesting the curve shown on the right.The graph becomes more steep as we move from left to right, suggesting that iPod sales were accelerating.Figure 3
23Functions from the Numerical, Algebraic, and Graphical Viewpoints From year 0 through year 3 the membership appears to increase more-or-less linearly (that is, the graph is almost a straight line), but then curves upward quite sharply from year 3 to year 6.This behavior can be modeled by using two different functions: one for the interval [0, 3] and another for the interval [3, 6] (see Figure 4).A function specified by two or more different formulas like this is called a piecewise-defined function.Figure 4
25Vertical Line TestEvery point in the graph of a function has the form (x, f (x)) for some x in the domain of f.Because f assigns a single value f (x) to each value of x in the domain, it follows that, in the graph of f, there should be only one y corresponding to any such value of x—namely, y = f (x).In other words, the graph of a function cannot contain two or more points with the same x-coordinate—that is, two or more points on the same vertical line.
26Vertical Line TestOn the other hand, a vertical line at a value of x not in the domain will not contain any points in the graph. This gives us the following rule.Vertical-Line TestFor a graph to be the graph of a function, every vertical line must intersect the graph in at most one point.
27Vertical Line Test Quick Examples As illustrated, only graph B passes the vertical line test, so only graph B is the graph of a function.
28Vertical Line TestTable 1 lists some common types of functions that are often used to model real world situations.A Compendium of Functions and Their GraphsTable 1
29Vertical Line Test cont’d A Compendium of Functions and Their Graphs Table 1
30Vertical Line Test cont’d A Compendium of Functions and Their Graphs Table 1
31Vertical Line Test cont’d A Compendium of Functions and Their Graphs Table 1
32Vertical Line Test cont’d A Compendium of Functions and Their Graphs Table 1
33Vertical Line TestFunctions and models other than linear ones are called nonlinear.