# Introduction to Functions Section 2.6

miL96478_ch02_191-202.qxd 9/19/09 5:10 AM Page 191 INTERM—AIE Section 2.6 Introduction to Functions Introduction to Functions Section 2.6 1. Definition of a Function Concepts In this section, we introduce a special type of relation called a function. 1. 2. 3. 4. DEFINITION Function Given a relation in x and y, we say “y is a function of x” if, for each element x in the domain, there is exactly one value of y in the range. Definition of a Function Vertical Line Test Function Notation Finding Function Values from a Graph 5. Domain of a Function Note: This means that no two ordered pairs may have the same first coordinate and different second coordinates. To understand the difference between a relation that is a function and a relation that is not a function, consider Example 1. Example 1 Determining Whether a Relation Is a Function Classroom Examples: p. 198, Exercises 6 and 10 Determine which of the relations define y as a function of x. a. b. x y 2 3 4 ⫺1 3 ⫺2 1 c. x y x 1 2 1 2 ⫺1 2 3 4 3 Solution: a. This relation is defined by the set of ordered pairs same x 511, 32, 11, 42, 12, ⫺12, 13, ⫺226 different y-values When x ⫽ 1, there are two possible range elements: y ⫽ 3 and y ⫽ 4. Therefore, this relation is not a function. b. This relation is defined by the set of ordered pairs 511, 42, 12, ⫺12, 13, 226. Notice that no two ordered pairs have the same value of x but different values of y. Therefore, this relation is a function. c. This relation is defined by the set of ordered pairs 511, 42, 12, 42, 13, 426. Notice that no two ordered pairs have the same value of x but different values of y. Therefore, this relation is a function. y 4 191 miL96478_ch02_191-202.qxd 9/19/09 5:11 AM Page 192 INTERM—AIE 192 Chapter 2 Linear Equations in Two Variables and Functions Instructor Note: Explain that a function can have the same y-coordinate with different x-coordinates, but not have the same x-coordinate with different y-coordinates. Skill Practice Determine if the relations define y as a function of x. 1. x y 2 12 6 13 7 10 2. {(4, 2), (⫺5, 4), (0, 0), (8, 4)} 3. {(⫺1, 6), (8, 9), (⫺1, 4), (⫺3, 10)} 2. Vertical Line Test y 5 4 3 2 1 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 ⫺1 ⫺2 Points align vertically 1 2 (4, 2) 3 4 5 x (4, ⫺2) ⫺3 ⫺4 ⫺5 Figure 2-32 A relation that is not a function has at least one domain element x paired with more than one range value y. For example, the set {(4, 2), (4, ⫺2)} does not define a function because two different y-values correspond to the same x. These two points are aligned vertically in the xy-plane, and a vertical line drawn through one point also intersects the other point (see Figure 2-32). If a vertical line drawn through a graph of a relation intersects the graph in more than one point, the relation cannot be a function. This idea is stated formally as the vertical line test. PROCEDURE The Vertical Line Test Consider a relation defined by a set of points (x, y) in a rectangular coordinate system. The graph defines y as a function of x if no vertical line intersects the graph in more than one point. The vertical line test can be demonstrated by graphing the ordered pairs from the relations in Example 1. a. 511, 32, 11, 42, 12, ⫺12, 13, ⫺226 b. 511, 42, 12, ⫺12, 13, 226 y y 5 5 Intersects more than once 4 3 2 4 3 2 1 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 ⫺1 ⫺2 1 1 3 4 5 ⫺3 ⫺4 ⫺5 Example 2 1. Yes 2. Yes 3. No 1 2 3 4 5 x ⫺5 Function No vertical line intersects more than once. Using the Vertical Line Test Use the vertical line test to determine whether the relations define y as a function of x. a. b. y y x Answers ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 ⫺1 ⫺2 ⫺3 ⫺4 Not a Function A vertical line intersects in more than one point. Classroom Examples: p. 199, Exercises 12 and 16 x x miL96478_ch02_191-202.qxd 7/21/09 12:11 PM Page 193 INTERM—AIE Section 2.6 Introduction to Functions 193 Solution: a. b. y y x x Not a Function A vertical line intersects in more than one point. Function No vertical line intersects in more than one point. Skill Practice Use the vertical line test to determine whether the relations define y as a function of x. 4. 5. y y x x 3. Function Notation A function is defined as a relation with the added restriction that each value in the domain must have only one corresponding y value in the range. In mathematics, functions are often given by rules or equations to define the relationship between two or more variables. For example, the equation y 2x defines the set of ordered pairs such that the y value is twice the x value. When a function is defined by an equation, we often use function notation. For example, the equation y 2x may be written in function notation as f 1x2 2x • • • f is the name of the function. x is an input value from the domain of the function. f (x) is the function value (y value) corresponding to x. The notation f (x) is read as “f of x” or “the value of the function f at x.” A function may be evaluated at different values of x by substituting x-values from the domain into the function. For example, to evaluate the function defined by f1x2 2x at x 5, substitute x 5 into the function. Avoiding Mistakes Be sure to note that f (x ) is not f ⴢ x. f 1x2 2x f 152 2152 f 152 10 TIP: f 152 10 can be written as the ordered pair (5, 10). Answers 4. Yes 5. No miL96478_ch02_191-202.qxd 7/21/09 12:11 PM Page 194 INTERM—AIE 194 Chapter 2 Linear Equations in Two Variables and Functions Thus, when x 5, the corresponding function value is 10. We say: f of 5 is 10. f at 5 is 10. f evaluated at 5 is 10. • • • The names of functions are often given by either lowercase or uppercase letters, such as f, g, h, p, K, and M. The input variable may also be a letter other than x. For example, the function y P(t) might represent population as a function of time. Evaluating a Function Example 3 Classroom Examples: p. 199, Exercises 18 and 22 Given the function defined by g1x2 12x 1, find the function values. a. g102 b. g122 c. g142 d. g122 Solution: 1 a. g1x2 x 1 2 g102 1 b. g1x2 x 1 2 1 102 1 2 g122 01 11 1 0 We say that “g of 0 is 1.” This is equivalent to the ordered pair 10, 12. We say that “g of 2 is 0.” This is equivalent to the ordered pair 12, 02. 1 c. g1x2 x 1 2 g142 d. 1 142 1 2 21 1 1 1 2 We say that “g of 2 is 2.” This is equivalent to the ordered pair 12, 22. Skill Practice Given the function defined by f (x) 2x 3, find the function values. y 5 4 g(x) 12 x 1 3 2 1 6. f (1) 7. f (0) 8. f (3) 1 9. f a b 2 (4, 1) 5 4 3 2 1 1 (2, 0) 1 (0, 1) 2 5 (2, 2) 4 5 Figure 2-33 Answers 7. 3 1 g1x2 x 1 2 1 g122 122 1 2 We say that “g of 4 is 1.” This is equivalent to the ordered pair 14, 12. 6. 5 1 122 1 2 8. 3 9. 4 x Notice that g102, g122, g142, and g122 correspond to the ordered pairs 10, 12, 12, 02, 14, 12, and 12, 22. In the graph, these points “line up.” The graph of all ordered pairs defined by this function is a line with a slope of 12 and y-intercept of 10, 12 (Figure 2-33). This should not be surprising because the function defined by g1x2 12x 1 is equivalent to y 12x 1. miL96478_ch02_191-202.qxd 9/19/09 5:12 AM Page 195 INTERM—AIE Section 2.6 Introduction to Functions Calculator Connections The values of g1x2 in Example 3 can be found using a Table feature. Y1 ⫽ 12x ⫺ 1 Function values can also be evaluated by using a Value (or Eval) feature. The value of g142 is shown here. A function may be evaluated at numerical values or at algebraic expressions, as shown in Example 4. Evaluating Functions Example 4 Given the functions defined by f1x2 ⫽ x ⫺ 2x and g1x2 ⫽ 3x ⫹ 5, find the function values. 2 b. g1w ⫹ 42 a. f1t2 Classroom Examples: p. 199, Exercises 24 and 38 c. f 1⫺t2 Solution: a. f1x2 ⫽ x2 ⫺ 2x f1t2 ⫽ 1t2 2 ⫺ 21t2 ⫽ t2 ⫺ 2t Substitute x ⫽ t for all values of x in the function. Simplify. g1x2 ⫽ 3x ⫹ 5 b. g1w ⫹ 42 ⫽ 31w ⫹ 42 ⫹ 5 Substitute x ⫽ w ⫹ 4 for all values of x in the function. ⫽ 3w ⫹ 12 ⫹ 5 ⫽ 3w ⫹ 17 c. f 1x2 ⫽ x2 ⫺ 2x f 1⫺t2 ⫽ 1⫺t2 ⫺ 21⫺t2 Simplify. Substitute ⫺t for x. 2 ⫽ t 2 ⫹ 2t Simplify. Skill Practice Given the function defined by g(x ) ⫽ 4x ⫺ 3, find the function values. 10. g(a) 11. g(x ⫹ h) 12. g (⫺x ) Answers 10. 4a ⫺ 3 12. ⫺4x ⫺ 3 11. 4x ⫹ 4h ⫺ 3 195 miL96478_ch02_191-202.qxd 7/21/09 12:12 PM Page 196 INTERM—AIE 196 Chapter 2 Linear Equations in Two Variables and Functions 4. Finding Function Values from a Graph We can find function values by looking at a graph of the function. The value of f(a) refers to the y-coordinate of a point with x-coordinate a. Classroom Example: p. 200, Exercise 58 Example 5 Finding Function Values from a Graph Consider the function pictured in Figure 2-34. h(x) 5 a. Find h112 . y h(x) 4 3 2 1 b. Find h122 . c. Find h152 . 5 4 3 2 1 1 2 d. For what value of x is h1x2 3? 1 2 3 4 5 x 3 e. For what values of x is h1x2 0? 4 5 Solution: Figure 2-34 a. h112 2 This corresponds to the ordered pair (1, 2). b. h122 1 This corresponds to the ordered pair (2, 1). c. h152 is not defined. The value 5 is not in the domain. d. h1x2 3 for x 4 This corresponds to the ordered pair (4, 3). e. h1x2 0 for x 3 and x 4 These are the ordered pairs (3, 0) and (4, 0). Skill Practice Refer to the function graphed here. 13. 14. 15. 16. 17. Find f (0). Find f (2). Find f (5). For what value(s) of x is f (x ) 0? For what value(s) of x is f (x ) 4? y 5 4 3 2 1 5 4 3 2 1 1 2 y f(x) 1 2 3 4 5 x 3 4 5 5. Domain of a Function A function is a relation, and it is often necessary to determine its domain and range. Consider a function defined by the equation y f 1x2 . The domain of f is the set of all x-values that when substituted into the function produce a real number. The range of f is the set of all y-values corresponding to the values of x in the domain. To find the domain of a function defined by y f 1x2, keep these guidelines in mind. • • Answers 13. 15. 16. 17. 3 14. 1 not defined x 4 and x 4 x 5 Exclude values of x that make the denominator of a fraction zero. Exclude values of x that make the expression within a square root negative. miL96478_ch02_191-202.qxd 9/19/09 5:21 AM Page 197 INTERM—AIE Section 2.6 Finding the Domain of a Function Example 6 Classroom Examples: p. 201, Exercises 74, 80, and 88 Write the domain in interval notation. a. f1x2 ⫽ x⫹7 2x ⫺ 1 c. k1t2 ⫽ 1t ⫹ 4 b. h1x2 ⫽ 197 Introduction to Functions x⫺4 x2 ⫹ 9 Instructor Note: Example 6 shows students that they can find the domain of a function without needing its graph. If you want to emphasize the graphical relationship, show the students the graphs of these functions in class. Or have the students graph the functions on a graphing calculator. d. g1t2 ⫽ t2 ⫺ 3t Solution: a. f 1x2 ⫽ 2xx ⫹⫺ 71 will not be a real number when the denominator is zero, that is, when 2x ⫺ 1 ⫽ 0 2x ⫽ 1 x⫽ 1 2 The value x ⫽ 12 must be excluded from the domain. 1 2 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 ((( 0 1 2 3 4 5 6 1 1 Interval notation: a⫺⬁, b ´ a , ⬁b 2 2 b. For h1x2 ⫽ xx2 ⫺⫹ 49 the quantity x2 is greater than or equal to 0 for all real numbers x, and the number 9 is positive. The sum x2 ⫹ 9 must be positive for all real numbers x. The denominator will never be zero; therefore, the domain is the set of all real numbers. Interval notation: 1⫺⬁, ⬁2 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 3 4 5 6 c. The function defined by k1t2 ⫽ 1t ⫹ 4 will not be a real number when the radicand is negative. The domain is the set of all t values that make the radicand greater than or equal to zero: t⫹4ⱖ0 t ⱖ ⫺4 Interval notation: 3⫺4, ⬁ 2 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 3 4 5 6 d. The function defined by g1t2 ⫽ t2 ⫺ 3t has no restrictions on its domain because any real number substituted for t will produce a real number. The domain is the set of all real numbers. Interval notation: 1⫺⬁, ⬁2 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 3 4 5 6 Skill Practice Write the domain in interval notation. 18. f 1x2 ⫽ 2x ⫹ 1 x⫺9 20. g 1x2 ⫽ 1x ⫺ 2 19. k 1x 2 ⫽ ⫺5 4x 2 ⫹ 1 21. h 1x2 ⫽ x ⫹ 6 Answers 18. 1⫺⬁, 92 ´ 19, ⬁2 20. 32, ⬁2 19. 1⫺⬁, ⬁ 2 21. 1⫺⬁, ⬁2 miL96478_ch02_191-202.qxd 8/10/09 8:18 PM Page 198 INTERM—AIE 198 Chapter 2 Linear Equations in Two Variables and Functions Section 2.6 Practice Exercises Boost your GRADE at ALEKS.com! • Practice Problems • Self-Tests • NetTutor For additional exercises, see Classroom Activities 2.6A–2.6D in the Instructor’s Resource Manual at www.mhhe.com/moh. • e-Professors • Videos Study Skills Exercises 1. Look back over your notes for this chapter. Have you highlighted the important topics? Have you underlined the key terms? Have you indicated the places where you are having trouble? If you find that you have problems with a particular topic, write a question that you can ask your instructor either in class or in the instructor’s office. 2. Define the key terms. a. Function b. Function notation c. Domain d. Range e. Vertical line test Review Exercises For Exercises 3–4, a. write the relation as a set of ordered pairs, b. identify the domain, and c. identify the range. 3. Parent, x Child, y Kevin Kayla Kevin Kira Kathleen Katie Kathleen y 4. a. {(⫺2,⫺4), (⫺1, ⫺1), (0, 0), (1, ⫺1), (2, ⫺4)} b. {⫺2, ⫺1, 0, 1, 2} c. {⫺4, ⫺1, 0} 5 4 3 2 1 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 ⫺1 ⫺2 Kira a. {(Kevin, Kayla), (Kevin, Kira), (Kathleen, Katie), (Kathleen, Kira)} b. {Kevin, Kathleen} c. {Kayla, Katie, Kira} 1 2 3 4 5 x ⫺3 ⫺4 ⫺5 Concept 1: Definition of a Function For Exercises 5–10, determine if the relation defines y as a function of x. (See Example 1.) 5. x 6. y x y ⫺2 1 ⫺2 1 2 2 2 2 0 8 0 8 7. u 21 w 23 x 24 y 25 z 26 Not a function Function Not a function 9. 511, 22, 13, 42, 15, 42, 1⫺9, 326 8. 3 10 4 Function 1 1 10. e10, ⫺1.12, a , 8b, 11.1, 82, a4, bf 2 2 Function 12 5 6 Function Writing Translating Expression Geometry Scientific Calculator Video miL96478_ch02_191-202.qxd 7/21/09 12:12 PM Page 199 INTERM—AIE Section 2.6 199 Introduction to Functions Concept 2: Vertical Line Test For Exercises 11–16, use the vertical line test to determine whether the relation defines y as a function of x. (See Example 2.) 11. 12. y 13. y x x x Function Not a function 14. y 15. y Function 16. y x y x Not a function x Not a function Not a function Concept 3: Function Notation Consider the functions defined by f1x2 6x 2, g1x2 x2 4x 1, Exercises 17–48, find the following. (See Examples 3–4.) h1x2 7, and k1x2 0 x 2 0 . For 17. g122 11 18. k122 0 19. g102 1 20. h102 7 21. k102 2 22. f 102 2 23. f 1t2 6t 2 24. g1a2 a2 4a 1 25. h1u2 7 26. k1v2 冟v 2冟 27. g132 29. k122 33. g12x2 30. f 162 4 4x2 8x 1 37. h1a b2 7 38 28. h152 4 31. f 1x 12 32. h1x 12 6x 4 36. g1a2 2 34. k1x 32 冟 x 5冟 35. g1p2 38. f 1x h2 6x 6h 2 39. f 1a2 6a 2 40. g1b2 1 1 44. g a b 4 p2 4p 1 41. k1c2 冟c 2冟 42. h1x2 7 1 43. f a b 2 1 45. h a b 7 7 3 46. k a b 2 1 2 47. f 12.82 7 18.8 7 a4 4a2 1 b2 4b 1 48. k15.42 1 16 7.4 Consider the functions p 51 12, 62, 12, 72, 11, 02, 13, 2p26 and q 516, 42, 12, 52, 1 34, 15 2, 10, 926. For Exercises 49–56, find the function values. 49. p122 7 50. p112 53. q122 5 3 54. qa b 4 Writing 0 1 5 Translating Expression 51. p132 2p 1 52. pa b 2 55. q162 4 56. q102 Geometry Scientific Calculator Video 6 9 miL96478_ch02_191-202.qxd 7/21/09 12:12 PM Page 200 INTERM—AIE 200 Chapter 2 Linear Equations in Two Variables and Functions Concept 4: Finding Function Values from a Graph 57. The graph of y f 1x2 is given. (See Example 5.) a. Find f102 . 3 b. Find f132. 1 c. Find f122. y 5 1 d. For what value(s) of x is f1x2 3? e. For what value(s) of x is f1x2 3? f. Write the domain of f. g. Write the range of f. 5 4 3 2 1 1 2 x 3 1 c. Find g142. 1 1, 34 x 3, x 5 f. Write the domain of g. g. Write the range of g. H142 is not defined because 4 is not in the domain of H. 4 5 4 3 2 1 5 4 3 2 1 1 2 3 d. For what value(s) of x is H1x2 3? x 3 and x 2 e. For what value(s) of x is H1x2 2? 3 4 5 x 4 5 3 4, 42 3 2, 52 60. The graph of y K1x2 is given. y 5 4 3 y K(x) 2 1 1 b. Find K152. K152 is not defined because 5 is not in the domain of K. 0 c. Find K112. d. For what value(s) of x is K1x2 0? x 1, x 1, and x 3 e. For what value(s) of x is K1x2 3? f. Write the domain of K. 2 x y y H(x) c. Find H132. g. Write the range of H. 1 4 5 1, 3 4 b. Find H142. f. Write the domain of H. 2 3 4 5 33, 2 3 All x on the interval 冤2, 1冥 1 2 3 59. The graph of y H1x2 is given. Writing x y g(x) 5 4 3 2 1 1 e. For what value(s) of x is g1x2 0? g. Write the range of K. 5 y 5 4 3 2 1 x2 a. Find K102. 3 4 1, 5 4 d. For what value(s) of x is g1x2 3? a. Find H132. 2 4 5 2 b. Find g112. 1 3 x 0, x 2 58. The graph of y g1x2 is given. a. Find g112. y f(x) 4 3 2 1 x 4 5 4 3 2 1 1 2 3 1 2 3 4 5 x 4 5 15, 5 4 冤1, 42 Translating Expression Geometry Scientific Calculator Video miL96478_ch02_191-202.qxd 9/19/09 5:36 AM Page 201 INTERM—AIE Section 2.6 201 Introduction to Functions For Exercises 61–70, refer to the functions y ⫽ f 1x2 and y ⫽ g1x2 , defined as follows: f ⫽ 51⫺3, 52, 1⫺7, ⫺32, 1⫺32, 42, 11.2, 526 g ⫽ 510, 62, 12, 62, 16, 02, 11, 026 5⫺3, ⫺7, ⫺32 , 1.26 61. Identify the domain of f. 56, 06 63. Identify the range of g. 64. Identify the domain of g. 65. For what value(s) of x is f(x) ⫽ 5? 50, 2, 6, 16 66. For what value(s) of x is f(x) ⫽ ⫺3? ⫺3 and 1.2 67. For what value(s) of x is g(x) ⫽ 0? 69. Find f (⫺7). 55, ⫺3, 46 62. Identify the range of f. 68. For what value(s) of x is g(x) ⫽ 6? 6 and 1 ⫺3 70. Find g(0). Concept 5: Domain of a Function 71. Explain how to determine the domain of the function defined by f 1x2 ⫽ ⫺7 0 and 2 6 x⫹6 . x ⫺ 2 The domain is the set of all real numbers for which the denominator is not zero. Set the denominator equal to zero, and solve the resulting equation. The solution(s) to the equation must be excluded from the domain. In this case, setting x ⫺ 2 ⫽ 0 indicates that x ⫽ 2 must be excluded from the domain. The domain is 1⫺⬁, 22 ´ 12, ⬁2 . The domain is the set of all 72. Explain how to determine the domain of the function defined by g1x2 ⫽ 1x ⫺ 3. real numbers for which x ⫺ 3 is nonnegative. Set the quantity x ⫺ 3 ⱖ 0 and solve the inequality. The solution set for the inequality is the domain of the function. Thus, the domain is 3 3, ⬁ 2 . For Exercises 73–88, find the domain. Write the answers in interval notation. (See Example 6.) 73. k1x2 ⫽ x⫺3 x⫹6 1⫺⬁, ⫺62 ´ 1⫺6, ⬁2 77. h1p2 ⫽ 1⫺⬁, ⬁ 2 p⫺4 p2 ⫹ 1 74. m1x2 ⫽ x⫺1 x⫺4 75. f1t2 ⫽ 1⫺⬁, 42 ´ 14, ⬁ 2 78. n1p2 ⫽ 1⫺⬁, ⬁ 2 5 t 76. g1t2 ⫽ 1⫺⬁, 02 ´ 10, ⬁ 2 p⫹8 1⫺⬁, 02 ´ 10, ⬁2 79. h1t2 ⫽ 1t ⫹ 7 p2 ⫹ 2 t⫺7 t 80. k1t2 ⫽ 1t ⫺ 5 3 ⫺7, ⬁ 2 35, ⬁2 81. f1a2 ⫽ 1a ⫺ 3 82. g1a2 ⫽ 1a ⫹ 2 83. m1x2 ⫽ 11 ⫺ 2x 84. n1x2 ⫽ 112 ⫺ 6x 85. p1t2 ⫽ 2t ⫹ t ⫺ 1 86. q1t2 ⫽ t ⫹ t ⫺ 1 87. f1x2 ⫽ x ⫹ 6 88. g1x2 ⫽ 8x ⫺ p 3 3, ⬁2 1⫺⬁, ⬁ 2 2 3 ⫺2, ⬁ 2 1⫺⬁, ⬁ 2 1⫺⬁, 12 4 3 1⫺⬁, 2 4 1⫺⬁, ⬁ 2 1⫺⬁, ⬁ 2 Mixed Exercises 89. The height (in feet) of a ball that is dropped from an 80-ft building is given by h1t2 ⫽ ⫺16t2 ⫹ 80, where t is the time in seconds after the ball is dropped. a. Find h(1) and h(1.5) h(1) ⫽ 64 and h(1.5) ⫽ 44 h(1) ⫽ 64 means that after 1 sec, the height of the ball is 64 ft. h(1.5) ⫽ 44 means that after 1.5 sec, the height of the ball is 44 ft. b. Interpret the meaning of the function values found in part (a). 90. A ball is dropped from a 50-m building. The height (in meters) after t sec is given by h1t2 ⫽ ⫺4.9t2 ⫹ 50. a. Find h(1) and h(1.5). h(1) ⫽ 45.1 and h(1.5) ⫽ 38.975 h(1) ⫽ 45.1 means that after 1 sec, the height of the ball is 45.1 m. h(1.5) ⫽ 38.975 means that after 1.5 sec, the height of the ball is 38.975 m. b. Interpret the meaning of the function values found in part (a). 91. If Alicia rides a bike at an average speed of 11.5 mph, the distance that she rides can be represented by d1t2 ⫽ 11.5t, where t is the time in hours. a. Find d(1) and d(1.5). d(1) ⫽ 11.5 and d(1.5) ⫽ 17.25 d(1) ⫽ 11.5 means that after 1 hr, the distance d(1.5) ⫽ 17.25 means that after 1.5 hr, the distance traveled is 17.25 mi. b. Interpret the meaning of the function values found in part (a). traveled is 11.5 mi. Writing Translating Expression Geometry Scientific Calculator Video miL96478_ch02_191-202.qxd 9/19/09 5:38 AM Page 202 INTERM—AIE 202 Chapter 2 Linear Equations in Two Variables and Functions 92. Brian’s score on an exam is a function of the number of hours he spends studying. The function defined by P1x2 100x2 (x 0) indicates that he will achieve a score of P% if he studies for x hours. 50 x2 Evaluate P(0), P(5), P(10), P(15), P(20), and P(25) and confirm the values on the graph. (Round to one decimal place.) Interpret P(25) in the context of this problem. Student Score (Percent) as a Function of Study Time D Percent P102 0% P152 ⬇ 33.3% P1102 ⬇ 66.7% P1152 ⬇ 81.8% P1202 ⬇ 88.9% P1252 ⬇ 92.6%. If Brian studies 25 hr he will get a score of 92.6%. P(x) 100 90 80 70 60 50 40 30 20 10 0 A 0 F E C B 5 10 15 20 Study Time (hr) 25 30 x For Exercises 93–96, write a function defined by y f(x) subject to the conditions given. 93. The value of f(x) is three more than two times x. 94. The value of f(x) is four less than the square of x. f (x) 2x 3 f(x) x2 4 95. The value of f(x) is ten less than the absolute value of x. f(x) |x| 10 96. The value of f(x) is sixteen times the square root of x. f(x) 16 1x Expanding Your Skills For Exercises 97–98, write the domain in interval notation. 97. q1x2 12, 2 2 98. p1x2 2x 2 14, 2 8 2x 4 Graphing Calculator Exercises 99. Graph k1t2 1t 5. Use the graph to support your answer to Exercise 80. Section 2.7 100. Graph h1t2 1t 7. Use the graph to support your answer to Exercise 79. Graphs of Functions Concepts 1. Linear and Constant Functions 1. Linear and Constant Functions 2. Graphs of Basic Functions 3. Definition of a Quadratic Function 4. Finding the x- and y-Intercepts of a Graph Defined by y ⫽ f (x) A function may be expressed as a mathematical equation that relates two or more variables. In this section, we will look at several elementary functions. We know from Section 2.1 that an equation in the form y k, where k is a constant, is a horizontal line. In function notation, this can be written as f1x2 k. For example, the graph of the function defined by f1x2 3 is a horizontal line, as shown in Figure 2-35. We say that a function defined by f1x2 k is a constant function because for any value of x, the function value is constant. Writing Translating Expression Geometry Scientific Calculator Video f(x) 5 f(x) 3 4 3 2 1 5 4 3 2 1 1 2 1 2 3 4 5 Figure 2-35 3 4 5 x

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