1. 1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 2 Graphs and Functions

2. OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Relations and Functions SECTION 2.4 1 2 3 4 Define relation. Define function. Find the domain of a function. Get information about a function from its graph. Solve applied problems by using functions. 5

3. DEFINTION OF A RELATION Any set of ordered pairs is called a relation.The set of all first components is called the domain of the relation, and the set of all second components is called the range of the relation. 3 © 2010 Pearson Education, Inc. All rights reserved

4. EXAMPLE 1 Finding the Domain & Range of a Relation Find the domain and range of the relation Solution The domain is the set of all first components, or {Titanic, Star Wars IV, Shrek 2, E.T., Star Wars I, Spider-Man}. { (Titanic, $600.8), (Star Wars IV, $461.0), (Shrek 2, $441.2), (E.T., $435.1), (Star Wars I, $431.1), (Spider-Man, $403.7)}. The range is the set of all second components, or {$600.8, $461.0, $441.2, $435.1, $431.1, $403.7)}. 4 © 2010 Pearson Education, Inc. All rights reserved

5. 5 Definitions A special relationship such as y = 10x in which to each element x in one set there corresponds a unique element y in another set is called a function. Because the value of y depends on the value of x, y is called the dependent variable and x is called the independent variable.

6. 6 © 2010 Pearson Education, Inc. All rights reserved DEFINTION OF A FUNCTION A function is a relation in which each element of the domain corresponds to one and only one element of the range.

7. 7 © 2010 Pearson Education, Inc. All rights reserved A relation may be described by a correspondence diagram: Definitions

8. 8 © 2010 Pearson Education, Inc. All rights reserved For each x in the domain of f, there corresponds a unique y in its range. The number y is denoted by f (x) read as “f of x” or “f at x”. We call f(x) the value of f at the number x and say that f assigns the f (x) value to x. FUNCTION NOTATION

9. 9 © 2010 Pearson Education, Inc. All rights reserved FUNCTIONS Tables and graphs can be used to describe functions. A table:

10. 10 © 2010 Pearson Education, Inc. All rights reserved FUNCTIONS A scatter diagram or a graph:

11. 11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Determining Whether an Equation Defines a Function Determine whether each equation determines y as a function of x. a. 6x 2 – 3y = 12 b. y 2 – x 2 = 4 One value of y corresponds to each value of x so it defines y as a function of x. Solution a.

12. 12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Determining Whether an Equation Defines a Function Solution continued b. y 2 – x 2 = 4 Two values of y correspond to each value of x so y is not a function of x.

13. 13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Evaluating a Function Solution Let g be the function defined by the equation y = x 2 – 6x + 8. Evaluate each function value.

14. 14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Evaluating a Function Solution continued

15. 15 © 2010 Pearson Education, Inc. All rights reserved AGREEMENT ON DOMAIN If the domain of a function that is defined by an equation is not explicitly specified, then we take the domain of the function to be the largest set of real numbers that result in real numbers as outputs.

16. 16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Domain of a Function Find the domain of each function. Solution a. f is not defined when the denominator is 0. Domain: {x|x ≠ –1 and x ≠ 1}

17. 17 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Domain of a Function Solution continued The square root of a negative number is not a real number and is excluded from the domain. Domain: {x|x ≥ 0}, [0, ∞)

18. 18 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Domain of a Function Solution continued The square root of a negative number is not a real number, so x – 1 ≥ 0 and since therefore denominator ≠ 0, x > 1. Domain: {x|x > 1}, or (1, ∞)

19. 19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Domain of a Function Solution continued Any real number substituted for t yields a unique real number. Domain: {t|t is a real number}, or (–∞, ∞)

20. 20 © 2010 Pearson Education, Inc. All rights reserved VERTICAL LINE TEST If no vertical line intersects the graph of a relation at more than one point, then the graph is the graph of a function. Graph does not represent a function

21. 21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Use the vertical-line test to determine which graphs are graphs of functions. Identifying the Graph of a Function Solution Not a function Does not pass the vertical line test since a vertical line can be drawn through the two points farthest to the left

22. 22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Use the vertical-line test to determine which graphs are graphs of functions. Identifying the Graph of a Function Solution Not a function Does not pass the vertical line test

23. 23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Use the vertical-line test to determine which graphs are graphs of functions. Identifying the Graph of a Function Solution Is a function Does pass the vertical line test

24. 24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Use the vertical-line test to determine which graphs are graphs of functions. Identifying the Graph of a Function Solution Is a function Does pass the vertical line test

25. 25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Let a.Is the point (1, –3) on the graph of f ? b.Find all values of x such that (x, 5) is on the graph of f. c.Find all y-intercepts of the graph of f. d.Find all x-intercepts of the graph of f. Examining the Graph of a Function Solution a. Check whether (1, –3) satisfies the equation. (1, –3) is not on the graph of f. ?

26. 26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 b.Find all values of x such that (x, 5) is on the graph of f. Substitute 5 for y and solve for x. Examining the Graph of a Function Solution continued (–2, 5) and (4, 5) are on the graph of f.

27. 27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 c.Find all y-intercepts of the graph of f. Substitute 0 for x and solve for y. Examining the Graph of a Function Solution continued The only y-intercept is –3.

28. 28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 d.Find all x-intercepts of the graph of f. Substitute 0 for y and solve for x. Examining the Graph of a Function Solution continued The x-intercepts of the graph of f are –1 and 3.

29. 29 © 2010 Pearson Education, Inc. All rights reserved FUNCTION INFORMATION FROM ITS GRAPH 1. Point on a graph A point (a, b) is on the graph of f means that a is in the domain of f and the value of f at a is b; that is, f(a) = b. We can visually determine whether a given point is on the graph of a function.

30. 30 © 2010 Pearson Education, Inc. All rights reserved FUNCTION INFORMATION FROM ITS GRAPH 2. Domain and range from a graph To determine the domain of a function, we look for the portion on the x-axis that is used in graphing f. We can find this portion by projecting (collapsing) the graph on the x-axis. This projection is the domain of f. The range of f is the projection of its graph on the y-axis.

31. 31 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Use the graphs to find the domain and the range of each function. Finding the Domain and Range from a Graph

32. 32 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 a. The open circle at (–2, 1) indicates that the point does not belong to the graph of f, while the full circle at the point (3, 3) indicates that the point is part of the graph. When we project the graph onto the x-axis, we obtain the interval (–2, 3] (the domain). Similarly, the projection of the graph of onto the y-axis gives the interval (1, 4] (the range). Finding the Domain and Range from a Graph Solution

33. 33 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 b. The projection of the graph onto the x-axis is made up of the two intervals [–3, 1] and [3, ∞]. So the domain of g is [–3, 1]  [3, ∞]. The projection of the line segment joining (−3, −1) and (1, 1) onto the y-axis is the interval [−1, 1]. The projection of the horizontal ray starting at the point (3, 4) onto the y-axis is just a single point at y = 4. Therefore, the range of g is [−1, 1]  {4}. Finding the Domain and Range from a Graph Solution continued

34. 34 © 2010 Pearson Education, Inc. All rights reserved FUNCTION INFORMATION FROM ITS GRAPH 3. Evaluations a. Finding f 1 c 2 Given a number c in the domain of f, we find f(c) from the graph of f.

35. 35 © 2010 Pearson Education, Inc. All rights reserved FUNCTION INFORMATION FROM ITS GRAPH 3. Evaluations b. Solving f 1 x 2 = d Given d in the range of f, find values of x for which f(x) = d.

36. 36 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 The figure on the next slide shows the graph of a function f that relates the unemployment rate, y, of a country with its GDP in billions of dollars. Here unemployment rate = percentage of people unemployed and GDP = Gross domestic product, the monetary value of all goods and services produced in the country. Obtaining Information from a Graph

37. 37 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Find and interpret. a.The domain of f b. The range of f c. f(30) d. f(x) = 40 Obtaining Information from a Graph

38. 38 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 a. Because the independent variable x denotes the GDP, we have x ≥ 0. The domain of f is the interval [0, ∞). This interval is the projection of the graph onto the x-axis. Obtaining Information from a Graph Solution b. The unemployment rate, y, is the percentage of people unemployed; so 0  y  100. The graph shows that as the GDP of the country increases, the unemployment rate drops but does not equal zero. So the range of f is (0, 100].

39. 39 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 c. From the graph, we find f(30) = 80. This means that if the country has $30 billion of GDP, its unemployment rate will be 80%. Obtaining Information from a Graph Solution continued d. From the graph, we find that 40 = f(60). This means that to have an unemployment rate of 40%, the GDP of the country must be $60 billion.

40. 40 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Many drugs used to lower high blood cholesterol levels are called statins and are very popular and widely prescribed. Bioavailability is the amount of a drug you have ingested that makes it into your bloodstream. Cholesterol–Reducing Drugs

41. 41 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 A statin with a bioavailability of 30% has been prescribed for Boris. Boris is to take 20 mg of this statin every day. During the same day, one-half of the statin is filtered out of the body. Find the maximum concentration of the statin in the bloodstream on each of the first ten days of using the drug, and graph the result. Cholesterol–Reducing Drugs

42. 42 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Since the statin has 30% bioavailability and Boris takes 20 mg per day, the maximum concentration in the bloodstream is 30% of 20 mg, or 20(0.3) = 6 mg from each day’s prescription. Because one-half of the statin is filtered out of the body each day, the daily maximum concentration is Cholesterol–Reducing Drugs Solution

43. 43 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Cholesterol–Reducing Drugs Solution continued DayMax Concentration 16.000 29.000 310.500 411.250 511.625 611.813 711.906 811.953 911.977 1011.988 Maximum concentration approaches 12 mg.

44. 44 © 2010 Pearson Education, Inc. All rights reserved FUNCTIONS IN ECONOMICS b is the fixed cost a (the cost of producing each item) is called the marginal cost. Linear Cost Function Average Cost

45. 45 © 2010 Pearson Education, Inc. All rights reserved FUNCTIONS IN ECONOMICS Suppose x items can be sold (demanded) at a price of p dollars per item. Then a linear demand function usually has the form The constants m, d, n, and k depend on the given situation. Linear Price–Demand Function

46. 46 © 2010 Pearson Education, Inc. All rights reserved FUNCTIONS IN ECONOMICS Revenue Function Profit Function Revenue = (Price per item) (Number of items sold) Profit = Revenue – Cost

47. 47 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 10 Metro Entertainment Co. spent $100,000 on production costs for its off-Broadway play Bride and Prejudice. Once the play runs, each performance costs $1000 and the revenue from each is $2400. Using x to represent the number of shows, Breaking Even a.write cost function C(x). b.write revenue function R(x). c.write profit function P(x). d.determine how many showings of Bride and Prejudice must be held for Metro to break even?

48. 48 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 10 Breaking Even Solution

49. 49 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 10 Breaking Even Solution continued d. Metro will break even when P(x) = 0. So 72 is the number of shows required for Metro to break even.