college alzebra assignmment

3

159

Functions

1,500 P

y

1,000

500

1970 1975 1980 1985 1990 1995 2000 2005 2010 0

Figure 1 Standard and Poor’s Index with dividends reinvested (credit “bull”: modification of work by Prayitno hadinata; credit “graph”: modification of work by measuringWorth)

Introduction Toward the end of the twentieth century, the values of stocks of Internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. Figure 1 tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1,100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many Internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.

Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties.

ChAPTeR OUTlIne

3.1 Functions and Function notation 3.2 domain and Range 3.3 Rates of Change and behavior of graphs 3.4 Composition of Functions 3.5 Transformation of Functions 3.6 Absolute value Functions 3.7 Inverse Functions

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

176 CHAPTER 3 fuNctioNs

3.1 SeCTIOn exeRCISeS

veRbAl

1. What is the difference between a relation and a function?

2. What is the difference between the input and the output of a function?

3. Why does the vertical line test tell us whether the graph of a relation represents a function?

4. How can you determine if a relation is a one-to-one function?

5. Why does the horizontal line test tell us whether the graph of a function is one-to-one?

AlgebRAIC

For the following exercises, determine whether the relation represents a function.

6. {(a, b), (c, d), (a, c)} 7. {(a, b),(b, c),(c, c)}

For the following exercises, determine whether the relation represents y as a function of x. 8. 5x + 2y = 10 9. y = x 2 10. x = y 2

11. 3x 2 + y = 14 12. 2x + y 2 = 6 13. y = −2x 2 + 40x

14. y = 1 __ x 15. x = 3y + 5

_ 7y − 1 16. x = √

— 1 − y 2

17. y = 3x + 5 ______ 7x − 1 18. x 2 + y 2 = 9 19. 2xy = 1

20. x = y 3 21. y = x 3 22. y = √ —

1 − x 2

23. x = ± √ —

1 − y 24. y = ± √ —

1 − x 25. y 2 = x 2

26. y 3 = x 2

For the following exercises, evaluate the function f at the indicated values f (−3), f (2), f (−a), −f (a), f (a + h).

27. f (x) = 2x − 5 28. f (x) = −5x 2 + 2x − 1 29. f (x) = √ —

2 − x + 5

30. f (x) = 6x − 1 ______ 5x + 2 31. f (x) = ∣ x − 1 ∣ − ∣ x + 1 ∣

32. Given the function g(x) = 5 − x 2, simplify g(x + h) − g(x)

__ h

, h ≠ 0

33. Given the function g(x) = x 2 + 2x, simplify g(x) − g(a)

_ x − a , x ≠ a

34. Given the function k(t) = 2t − 1: a. Evaluate k(2). b. Solve k(t) = 7.

35. Given the function f (x) = 8 − 3x: a. Evaluate f (−2). b. Solve f (x) = −1.

36. Given the function p(c) = c 2 + c: a. Evaluate p(−3). b. Solve p(c) = 2.

37. Given the function f (x) = x 2 − 3x a. Evaluate f (5). b. Solve f (x) = 4

38. Given the function f (x) =  √ —

x + 2 : a. Evaluate f (7). b. Solve f (x) = 4

39. Consider the relationship 3r + 2t = 18. a. Write the relationship as a function r = f (t). b. Evaluate f (−3). c. Solve f (t) = 2.

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SECTION 3.1 sectioN exercises 177

gRAPhICAl

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

40.

x

y 41.

x

y 42.

x

y

43.

x

y 44. 45.

x

y

46.

x

y 47.

x

y 48.

x

y

49.

x

y 50.

x

y 51.

x

y

x

y

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

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178 CHAPTER 3 fuNctioNs

52. Given the following graph a. Evaluate f (−1). b. Solve for f (x) = 3.

53. Given the following graph a. Evaluate f (0). b. Solve for f (x) = −3.

54. Given the following graph a. Evaluate f (4). b. Solve for f (x) = 1.

For the following exercises, determine if the given graph is a one-to-one function. 55.

x

y 56.

x

y 57.

x

y

58.

x

y 59.

x

y

π�π

5 4 3 2 1

�1 �2 �3 �4 �5

nUmeRIC For the following exercises, determine whether the relation represents a function.

60. {(−1, −1),(−2, −2),(−3, −3)} 61. {(3, 4),(4, 5),(5, 6)} 62. {(2, 5),(7, 11),(15, 8),(7, 9)}

For the following exercises, determine if the relation represented in table form represents y as a function of x.

63. x 5 10 15 y 3 8 14

64. x 5 10 15 y 3 8 8

65. x 5 10 10 y 3 8 14

For the following exercises, use the function f represented in Table 14 below.

x 0 1 2 3 4 5 6 7 8 9 f (x) 74 28 1 53 56 3 36 45 14 47

Table 14

66. Evaluate f (3). 67. Solve f (x) = 1

x

 
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