Math 2412 Activity 1(Due by EOC Sep. 17)

Transcription

1 Math 4 Activity (Due by EOC Sep. 7) Determine whether each relation is a unction.(indicate why or why not.) Find the domain and range o each relation.. 4,5, 6,7, 8,8. 5,6, 5,7, 6,6, 6,7 Determine whether each equation deines y as a unction o. Show why or why not.. y 5 4. y 5 5. y 0 Evaluate each unction at the given values o the independent variable. Simpliy a) 6 b) 4 7. a) 5 b) 5 4 c) 5 c) Use the graph o to solve problems 8-0. Graph o d) 6 d) Solve. 5. Solve Solve 7. Domain o 8. Range o.

2 9. -intercept(s) 0. y-intercept. I is a real-valued unction with the property that y y or all real numbers and y, and i, then ind a) {Hint:.} b) {Hint:.} c) 4 {Hint: 4 d),000,000.} e) {Hint:.} ) 0 {Hint: 0 0. } g) {Hint: 0. }. I is a real-valued unction with the property that y y or all real numbers and y, and i, then ind a) {Hint:.} b) {Hint:.} c) 4 {Hint: 4 d),000.} e) Show that 0 or all real numbers,. {Hint:.} ) {Hint:.} g) 0 {Hint: 0 0. }

3 Evaluate the ollowing unctions at the given value o the independent variable. Simpliy.. 6; 0 7 ; 0 4. g a) a) g b) 0 b) g 5 c) c) g6 5 ; ; 5 Use the graph o the unction on the interval,, which consists o points and line segments, to answer the ollowing questions.(5-4) What s the domain o? 6. What s the range o? 7. What are the -intercepts? 8. Where is increasing? 9. Where is decreasing? 0. Where is constant?. Where does have a local maimum?. Where does have a local minimum?. Solve. 4. Solve Determine i the ollowing unctions are even, odd, or neither.(5-4) Justiy your answer g 5 7. h 8. k 5 7

4 9. l ; 0 ; I is an even unction, and g is an odd unction, then determine i the ollowing unctions are even, odd, both, or neither. (4-45) Justiy your answer. 4. h g 4. j g 44. k 45. k g g Use the given graph o the unction, which consists o line segments, to graph the given unction. (46-59) 46. g 47. g

5 48. g 49. g 50. g 5. g 5. g 5. g 54. g 55. g 56. g 57. g 58. g 59. g Find the domains o the ollowing unctions.(60-65) g 6. h j 64. k 65. l 6 Using the graphs o the unctions and g, determine the ollowing: (66-77) Graph o Graph o g g 66. g 67. g 68. g 69.

6 g g 7. g 0 7. g g 74. g g 75. Solve g. 76. Solve g Solve g. 78. Find g, g, g, g, and g or,,,4,,, 4,,,,5,,0, 4, g. g g g g and g Graph the ollowing piecewise unctions ; 0 47 ; 0 5 ; 0 g ;0 ; ; 0 h ;0 ; h ; ; ; ;

7 8. Given that and or all, ind 08. {Hint: First ind 4, 7, 0,., and look or a pattern.} 84. For 5 9, or what values o is it true that? 85. For, or what values o is it true that 4? 86. Let F be a real-valued unction deined or all real ecept or 0 and and satisying the equation F F. Find all unctions F satisying these conditions. {Hint: Substitute and original equation.} into the equation,eliminate F, and use it with the 87. I and g, does g? Justiy your answer. 88. Determine the unction F which satisies the equation real. F F 4 or all {Hint: Substitute or in the equation and solve the resulting system or F.} 89. Suppose that H is a unction and is a number with H H a) What is H H H H? 80 times b) What is c) I H H H H H H? 8 times H, then what s H H H H? 80 times.

8 90. Given the graphs o the two unctions and g, sketch the graphs o the unctions g and g. g 9. Show that there do not eist unctions and g so that g y y or all and y. {Hint: 0 g 0 0 g 0 0.} 0 g 0 g 9. Show that there do not eist unctions and g so that g y y or all and y. {Hint: 0 g 0 0 g 0.} 0 g

9 9. Consider the unction ; 5 ; 5. ; 5 a) Find 6. b) Find 7. c) Find 8. d) Find 9. e) Find 5. ) Find 6. g) Find 7. h) Find 8 i) What s 6? j) Try to complete the graph on the interval 0,0..

10 94. Consider the unction 0 ; 00. Evaluate each o the ollowing: ; 00 0, 80, 97, 0, Find p 6 i p p p p. p is a unction or which ;. p, p, p, and I g and g, then ind. 97. The unction is deined by value o k i k 4, where k is a constant and. What is the. or all real numbers ecept 98. Suppose that a and b are real numbers with a b. Which o the ollowing inequalities must be true? I true, show why. I alse, give a speciic countereample. a) a 4 b 5 b) a b a b c) a b d) e) a b ab b 99. Find eamples o constant unctions that make the ollowing statements alse: a) b) 00. Let. Compute c) y y a) b) c) d) 0. Let. Compute a) b) c) 0. Let ; 0. Compute ; 0 a) b) c) d) e)

11 0. I is both even and odd, then ind a ormula or. 04. Let be a unction whose domain is symmetric about the origin, i.e. belongs to the domain i belongs to the domain. Show that is the sum o an even unction and an odd unction, E O. Consider and 05. Show that the decomposition o into an even part and an odd part is unique. That is show E O E O. This would imply that that i and E E O O, so use problem Write each o the ollowing unctions as the sum o an odd and an even unction: a) b) c) d) 07. I the domain o includes the interval,, then show that the unction cos even unction. What about sin? 08. Let be a unction whose domain is 0,. I g, is g odd or even? 09. I is an odd unction and g is an odd unction, what can you say about g? 0. I is an odd unction and g is an even unction, a) What can you say about g? b) What can you say about g?. I ; ;. is an and g, then ind all values o so that g. What can you say about a linear unction m b i: a) or all? b) c) or all? d) or all?? e) and? ) 4 and?.

12 . Given the complete graph o the unction, answer the ollowing questions: a) What is the domain o? b) What is the range o? c) Find. d) Find. e) Find. ) Find. Given the complete graphs o the unctions (rom the previous problem) and g, work the ollowing problems: (4-) 4. g0 5. g 6. g 7. g

13 8. g 9. g 0. g. g. Graph g.. Graph g. 4. Given the graph o the unction on the interval 5,5, graph the indicated unctions: a) g b) h 5. Find the domain o the unction. 6. I and g 7. I and g 8. I and g, then show that g., then show that g 9. I q and value o p.., then show that g p q , then ind the 0. Find two dierent numbers so that the square o each number is the other number.. For which three values o m will the equation solution? m m 4 0 have eactly one {Hint: It s not always a quadratic equation.}. Solve ,consider the cases o and. } {Hint: 4 4

14 . Solve. {Hint: You can break it into the two equations ; 0 and ; 0.} 4. Solve 4 0. {Hint: You can break it into the two equations 5. Solve. 4 0; 0 and 4 0; 0.} {Hint: You can break it into the three equations ;,, and ; ;.} 6. Suppose that is a non-decreasing unction deined on 0, with the properties that and or all in 0,. a) Find 0 and. b) Find and {Hint: 0 and ?.} {Hint:? and.} {Since is non-decreasing, the graph o must be a horizontal line segment or in 8 7 c) Find,,, and {Since is non-decreasing, the graph o must be horizontal line segments or in 7 8 and,.} d) Find,,,,,,, and ,.} {Since is non-decreasing, the graph o must be horizontal line segments or in ,,,,,, and ,.} 7 7 The unction is a amous unction called the Cantor Ternary Function, and its graph is called the devil s staircase. The ollowing is an approimate graph o using the unction values that you have determined, so ar. The ull graph is an eample o a sel-similar ractal., 9 9

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