StewartClegg_01_01.ppt

CHAPTER 1 Functions and Models Copyright © Cengage Learning. All rights reserved. 1.1 Functions and Their Representations Copyright © Cengage Learning. All rights reserved. Introduction to Functions 3 Introduction to Functions Mathematical relationships can be observed in virtually every a...

CHAPTER 1 Functions and Models Copyright © Cengage Learning. All rights reserved. 1.1 Functions and Their Representations Copyright © Cengage Learning. All rights reserved. Introduction to Functions 3 Introduction to Functions Mathematical relationships can be observed in virtually every aspect of our environment and daily lives. Populations, financial markets, the spread of diseases, setting the price of a new product, and the effects of pollution on an ecosystem can all be analyzed using mathematics. Many mathematical relationships can be considered as functions. A function is a correspondence in which one quantity is determined by another. 4 Introduction to Functions For instance, each day that the US stock market is open corresponds to a closing price of Google stock. We say that the daily closing price of the stock is a function of the date. We typically refer to a function by a single letter such as f. If x represents an input to the function f, the corresponding output is f (x), read “f of x.” 5 Introduction to Functions The set of all allowable inputs is called the domain of the function. The range of is the set of all possible output values, f (x), as varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable. 6 Introduction to Functions It’s helpful to think of a function as a machine (see Figure 2). Machine diagram for a function ƒ Figure 2 If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f (x) according to the rule of the function. 7 Example 1 – A Price Function A cafe sells its basic coffee in three different cup sizes: 8, 10, and 14 ounces. They charge $0.22 per ounce for the drinks. (a) If the function p is defined so that p (v) is the price of v ounces of coffee, find and interpret the value of p (10). (b) What are the domain and range of p? Solution: (a) The function value p (10) represents the output (price) of the function when the input is 10 ounces of coffee. 8 Example 1 – Solution cont’d Thus p (10) = $0.22  10 = $2.20 (b) If we assume that the cafe sells only 8-, 10-, and 14-ounce coffee drinks, then the only allowable inputs to the price function are the three numbers 8, 10, and 14, so the domain of p is the set {8, 10, 14}. The range is the set of outputs that correspond to the inputs in the domain: {1.76, 2.20, 3.08}. 9 Introduction to Functions The most common method for visualizing a function is to view its graph. If f is a function, then its graph is the set of input-output pairs (x, f (x)) plotted as points for all x in the domain of f. In other words, the graph of f consists of all points (x, y) in the coordinate plane such that y = f (x) and x is in the domain of f. If the domain consists of isolated values, as in Example 1, the data are discrete and the graph is a collection of individual points, called a scatter plot. 10 Introduction to Functions On the other hand, if the input variable represents a quantity that can vary continuously through an interval of values, the graph is a curve or line (see Figure 3). Graphs of functions Figure 3 11 Example 2 – Reading Information from a Graph The graph of a function f is shown in Figure 6. Figure 6 (a) Find the values of f (1) and f (5). (b) What are the domain and range of f ? 12 Example 2 – Solution (a) We see from Figure 6 that the point (1, 3) lies on the graph of f, so the value of f at 1 is f (1) = 3. (In other words, the point on the graph that lies above x = 1 is 3 units above the x-axis.) When x = 5, the graph lies about 0.7 unit below the x-axis, so we estimate that f (5)  –0.7. (b) We see that f (x) is defined when 0  x  7, so the domain of f is the closed interval [0, 7]. Notice that f takes on all values from –2 to 4, so the range of f is { y | –2  y  4} = [–2, 4] 13 Representations of Functions 14 Representations of Functions We have seen four possible ways to represent a function:  verbally  numerically  visually  algebraically (by a description in words) (by a table of values) (by a graph) (by an explicit formula) If a single function can be represented in several ways, it is often useful to go from one representation to another to gain additional insight into the function. But certain functions are described more naturally by one method than by another. 15 Example 3 – Drawing a Graph from a Verbal Description When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on. Solution: The initial temperature of the running water is close to room temperature because the water has been sitting in the pipes. When the water from the hot-water tank starts flowing from the faucet, T increases quickly. 16 Example 3 – Solution cont’d In the next phase, T is constant at the temperature of the heated water in the tank. When the tank is drained, T decreases to the temperature of the water supply. This enables us to make the rough sketch of T as a function of t in Figure 9. Figure 9 17 Representations of Functions A more accurate graph of the function in Example 3 could be obtained by using a thermometer to measure the temperature of the water at 10-second intervals. In general, researchers collect experimental data and use them to sketch the graphs of functions, as the next example illustrates. 18 Example 4 – A Numerically Defined Function The data shown in the margin give weekly sales figures for a video game shortly after its release. Let N (t) be the number of copies sold, in thousands, during the week ending weeks after the game’s release. Sketch a scatter plot of these data, and use the scatter plot to draw a continuous approximation to the graph of N (t). Then use the graph to estimate the number of copies sold during the sixth week. 19 Example 4 – Solution We plot the five points corresponding to the data from the table in Figure 10. Figure 10 20 Example 4 – Solution cont’d The data points in Figure 10 look quite well behaved, so we simply draw a smooth curve through them by hand as in Figure 11. Figure 11 21 Example 4 – Solution cont’d From the graph, it appears that N (6)  12.5, so we estimate that 12,500 units were sold during the sixth week. 22 Example 5 – Expressing a Cost as a Function A rectangular storage container with an open top has a volume of 10m3. The length of its base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the width of the base. 23 Example 5 – Solution We draw a diagram as in Figure 12 and introduce notation by letting w and 2w be the width and length of the base, respectively, and h be the height. The area of the base is (2w)w = 2w2, so the cost, in dollars, of the material for the base is 10(2w2). Figure 12 Two of the sides have area wh and the other two have area 2wh, so the cost of the material for the sides is 6[2(wh) + 2(2wh)]. 24 Example 5 – Solution cont’d The total cost is therefore C = 10(2w)2 + 6[2(wh) + 2(2wh)] = 20w2 + 36wh To express C as a function of w alone, we need to eliminate h and we do so by using the fact that the volume is 10m3. Thus volume = width  length  height = w(2w)h = 10 25 Example 5 – Solution cont’d which gives Substituting this into the expression for C, we have C = 20w2 + 36w = Therefore the equation C (w) = 20w2 + w>0 expresses C as a function of w. 26 Example 6 – A Function Defined by a Formula If f (x) = 2x2 – 5x + 1, evaluate (a) f (–3) (b) f (4) – f (2) (c) (h  0) 27 Example 6 – Solution (a) Replace x by –3 in the expression for f (x): f(–3) = 2(–3)2 – 5(–3) + 1 = 2  9 + 15 + 1 = 18 + 15 + 1 = 34 28 Example 6 – Solution cont’d (b) f (4) – f (2) = [2(4)2 – 5(4) + 1] – [2(2)2 – 5(2) + 1] = 13 – (–1) = 14 (c) We first evaluate f (1 + h) by replacing by 1 + h in the expression for f (x): f (1 + h) = 2(1 + h)2 – 5(1 + h) + 1 = 2(1 + 2h + h2) – 5(1 + h) + 1 29 Example 6 – Solution cont’d = 2 + 4h + 2h2 – 5 – 5h + 1 = 2h2 – h – 2 Then we substitute into the given expression and simplify: 30 Example 7 – Determining the Domain of a Function Defined by a Formula Find the domain of each function. (a) (b) Solution: (a) Because the square root of a negative number is not defined (as a real number), the domain of B consists of all values of r such that r + 2  0. This is equivalent to r  –2, so the domain is the interval [–2, ). 31 Example 7 – Solution cont’d (b) Since and division by 0 is not allowed, we see that g (x) is not defined when x = 0 or x = 1. Thus the domain of g is {x | x ≠ 0, x ≠ 1}. 32 Representations of Functions The graph of a function is a curve or scatter plot in the xy-plane. But the question arises: Which graphs in the xy-plan represent functions and which do not? This is answered by the following test. 33 Representations of Functions The reason for the truth of the Vertical Line Test can be seen in Figure 13. Figure 13 34 Representations of Functions If each vertical line x = a intersects a curve only once, at (a, b), then exactly one functional value is defined by f (a) = b. But if a line x = a intersects the curve twice, at (a, b) and (a, c), then the curve can’t represent a function because a function can’t assign two different output values to an input a. 35 Example 8 – Using the Vertical Line Test Determine whether the graph represents a function. (a) Figure 14 (b) Figure 15 36 Example 8 – Solution (a) Notice that if we draw a vertical line on the scatter plot in Figure 14 at or at x = –1 or at x = 2, the line will intersect two of the points. Therefore the scatter plot does not represent a function. (b) No matter where we draw a vertical line on the graph in Figure 15, the line will intersect the graph at most once, so this is the graph of a function. Notice that the “gap” in the graph does not pose any trouble; it is acceptable for a vertical line not to intersect the graph at all. 37 Mathematical Modeling 38 Mathematical Modeling A mathematical model is a mathematical description (usually by means of a function or an equation) of a real-world scenario, such as the demand for a company’s product or the life expectancy of a person at birth. Although a function used as a model may not exactly match observed data, it should be a close enough approximation to allow us to understand and analyze the situation, and perhaps to make predictions about future behavior. 39 Mathematical Modeling Figure 16 illustrates the process of mathematical modeling. Given a real-world problem, our first task is to formulate a mathematical model by identifying and naming the independent and dependent variables and making assumptions that simplify the situation enough to make it mathematically tractable. 40 Mathematical Modeling The second stage is to apply the mathematics that we know (such as the calculus that will be developed throughout this book) to the mathematical model that we have formulated in order to derive mathematical conclusions. Then, in the third stage, we take those mathematical conclusions and interpret them as information about the original real-world situation by way of offering explanations or making predictions. 41 Mathematical Modeling The final step is to test our predictions by checking against new real data. If the predictions don’t compare well with reality, we need to refine our model or formulate a new model and start the cycle again. 42 Piecewise Defined Functions 43 Piecewise Defined Functions In some instances, no single formula adequately describes the behavior of a quantity. A population may exhibit one growth pattern for 20 years but then change to a different trend. In such cases we can use a function with different formulas in different parts of the domain. We call such functions piecewise defined functions. 44 Example 9 – Graphing a Piecewise Defined Function A function f is defined by Evaluate f (–2), f (–1), and f (1) and sketch the graph. Solution: Remember that a function is a rule. For this particular function the rule is the following: First look at the value of the input x. If it happens that x  –1, then the value of f (x) is 1 – x. 45 Example 9 – Solution cont’d On the other hand, if x > –1, then the value of f (x) is x2. Since –2  –1, we have f (–2) = 1 – (–2) = 3. Since –1  –1, we have f (–1) = 1 – (–1) = 2. Since 1 > –1, we have f (1) = 12 = 1. 46 Piecewise Defined Functions How do we draw the graph of f ? We observe that if x  –1, then f (x) = 1 – x, so the part of the graph of f that lies to the left of x = –1 must coincide with the line y = 1 – x, which has slope –1 and y-intercept 1. If x > –1, then f (x) = x2, so the part of the graph of that lies to the right of the line x = –1 must coincide with the graph of y = x2, which is a parabola. 47 Piecewise Defined Functions This enables us to sketch the graph in Figure 17. Figure 17 The solid dot indicates that the point (–1, 2) is included on the graph; the open dot indicates that the point (–1, 1) is excluded from the graph. 48 Symmetry 49 Symmetry If a function f satisfies f (–x) = f (x) for every number x in its domain, then f is called an even function. For instance, the function f (x) = x2 is even because f (–x) = (–x)2 = x2 = f (x) The geometric significance of an even function is that its graph is symmetric with respect to the y-axis (see Figure 19). An even function Figure 19 50 Symmetry This means that if we have plotted the graph of f for x  0, we obtain the entire graph simply by reflecting this portion about the y-axis. If f satisfies f (–x) = –f (x) for every number x in its domain, then f is called an odd function. For example, the function f (x) = x3 is odd because f (–x) = (–x)3 = –x3 = –f (x) 51 Symmetry The graph of an odd function is symmetric about the origin (see Figure 20). An odd function Figure 20 52 Example 11 – Testing for Symmetry Determine whether each of the following functions is even, odd, or neither even nor odd. (a) f (x) = x5 + x (b) g (x) = 1 – x4 (c) h (x) = 2x – x2 Solution: (a) f (–x) = (–x)5 + (–x) = (–1)5x5 + (–x) = –x5 – x = –(x5 + x) = –f (x) Therefore f is an odd function. 53 Example 11 – Solution cont’d (b) g (–x) = 1 – (–x)4 = 1 – x4 = g (x) So g is even. (c) h (–x) = 2(–x) – (–x)2 = –2x – x2 Since h (–x) ≠ h (x) and h (–x) ≠ –h (x), we conclude that h is neither even nor odd. 54 Symmetry The graphs of the functions in Example 11 are shown in Figure 21. (a) Odd function (b) Even function (c) Neither even nor odd Figure 21 Notice that the graph of is symmetric neither about the y-axis nor about the origin. 55 

Show More Show Less