MATH 1306 Handout #1 Review of Functions

Transcription

1 Revised: Summer 2010 MATH 1306 Handout #1 Review of Functions 1. For each of the following functions, find the C-intercept and the B-intercept(s). 1a. 0ÐBÑ &B "! ii) B-intercept(s) # 1b. 1ÐBÑ B #B "& a.! b. "& c. # d. none ii) B-intercept(s) a. $ ß & b. $ ß & c. $ß & d. $ß & # 1c. C B 'B "" ii) B-intercept(s) % $ 1d. 2ÐBÑ B %B ii) B-intercept(s) # % 1e. 0ÐBÑ )B #B ii) B-intercept(s)

2 %Î$ 1f. C $B $B ii) B -intercept(s) ( Hint: Factor out $B.) 1g. 0ÐBÑ #ÐB "ÑÐB "ÑÐB $Ñ a. # b. $ c. " d. ' ii) B-intercept(s) a. "ß "ß $ b. #ß "ß "ß $ c. ' d. #ß "ß $ 2. Find the C-intercept, B-intercept(s), and domain of each of the following functions. 2a. 2ÐBÑ $ #B C-intercept B-intercept(s) domain # 2b. TÐBÑ B #B "& C-intercept B-intercept(s) domain Page 2

3 2c. 1ÐBÑ $B ' B " C-intercept a. " b. # c. ' d. $ B-intercept(s) a. " b. # c. ' d. $ domain a. all real numbers except! b. all real numbers except # and ' c. all real numbers d. all real numbers except " 2d. 7ÐBÑ ÐB #Ñ "Î# C-intercept a. none b. " c.! d. # B-intercept(s) a. none b. # c.! d. % domain a. all real numbers b. all real numbers except 2 c. all real numbers greater than or equal $ d. all real numbers at least # Page 3

4 2e. 8ÐBÑ ÐB #Ñ # C-intercept a. none b. " % c.! d. % B-intercept(s) a. none b. # c.! d. " domain a. all real numbers b. all real numbers except 2 c. all real numbers except! d. all real numbers at least # 3. A function named 0 is graphed here. Each of the following parts 3a 3c pertains to this function. 3a. Write the domain of 0. 3b. Find the B-intercept(s) of 0. 3c. Find the C-intercept of 0. Page 4

5 4. A function named 1 is graphed here. Each of the following parts 4a 4b pertains to this function. 4a. Find the B-intercept(s). 4b. Find the C-intercept. 5. The NASDAQ Composite Index is a numerical value that roughly measures the price of all stocks traded on the NASDAQ Stock Exchange. This exchange has become more important in recent years, since the stocks of many high-tech companies are traded there. The following graph depicts the value of the NASDAQ index ( M) over the course of a recent trading day. The input axis is marked off in military time, where > *Þ& corresponds to 9:30 am and >"' corresponds to 4 pm. 5a. Write the domain of the function M. In practical terms, what does this domain tell you about the NASDAQ Stock Exchange? 5b. Is the value ")$& in the range of the function M? Yes No 5c. Interpret MÐ"'Ñ in the problem context using a complete sentence. Page 5

6 5d. Evaluate MÐ"%Ñ 5e. Express the following statement in function notation: The value of the NASDAQ index at opening time was ")&!. 5f. Write the input and output variables. 5g. Write the units of measure for the input and output quantities. 6. The Hard-Body Elliptical Crosstrainer fitness machine has an LCD display that graphs the total calories burned during a workout. Suppose the display appeared roughly as shown below after Imani's last workout on the machine. Let X be the total calories burned after 7 minutes of the workout. 6a. Write the domain of the function X. a. all real numbers between! and "!, inclusive b. all real numbers between! and "'&, inclusive c. all real numbers d. all real numbers at least! In practical terms, what does this domain tell you about Imani's last workout? a. Imani burned "(! calories during her last workout. b. Imani burned "! calories during her last workout. c. Imani's last workout lasted "! minutes. d. The number of calories Imani burned increased steadily during her last workout. 6b. Is the value in the range of the function? Yes No "%! X a. Yes b. No Page 6

7 6c. Interpret XÐ%Ñin the problem context using a complete sentence. a. The number of minutes it took Imani to burn % calories. b. The number of calories Imani burned after % minutes. c. The number of calories Imani burned times %. d. Imani burned % calories per minute. 6d. Evaluate X Ð)Ñ a. "&! b. ')! c. " d. )! 6e. Express the following statement in function notation: Imani burned "(! calories altogether during her last workout. a. X "(! b. X Î"! "( c. X Ð"!Ñ "(! d. X Ð"(!Ñ "! 6f. Write the input and output variables. a. input p7, outputpx b. input pb, outputpc c. input pc, outputpb d. input px, outputp7 6g. Write the units of measure for the input and output quantities. a. inputpcalories, outputpminutes b. inputptime, outputpenergy c. inputpspeed, outputppounds d. inputpminutes, outputpcalories Page 7

8 7. The following graph shows the average number of cars (in thousands) traveling on the Gridlock Expressway in Gotham City during a typical weekday from ' am to ) pm (the input axis is marked off in military time). For example, military time "% corresponds to # pm. Suppose G is the number of cars (in thousands) on this expressway at time > (military time). 7a. Write the domain of the function G. 7b. Interpret GÐ"$Ñ in the problem context using a complete sentence. 7c. Evaluate GÐ#!Ñ. 7d. Express the following statement in function notation: There are "!ß!!! cars on the Gridlock Expressway at ) am on a typical weekday. 7e. Write the input and output variables. 7f. Write the units of measure for the input and output quantities. Page 8

9 8. The function Z 0Ð>Ñ graphed here depicts the number of hours each day typical Americans from age $ years to age '! years spend watching TV. 8a. Write the domain of the function 0. 8b. Is the value % in the range of the function 0? Yes No 8c. Interpret 0Ð%&Ñ in the problem context using a complete sentence. 8d. Evaluate 0Ð"!Ñ 8e. Express the following statement in function notation: A typical &! year-old American watches # hours of TV each day. 8f. Write the input and output variables. 8g. Write the units of measure for the input and output quantities. Page 9

10 9. A large tank of water is being drained. The amount of water ÐEÑ in the tank, in gallons, after > minutes have elapsed is given by the following function. EÐ>Ñ #!! "!> 9a. Find the value of the C-intercept and interpret its meaning. 9b. Find the value of the B-intercept and interpret its meaning. 10. A recording company produces compact disks (CDs). The company's profit, in dollars, for making and selling B CDs is given by the following function. T ÐBÑ #Þ&!B #%!!! The B-intercepts of the profit function are called break-even points because the company must make and sell this many CDs to break even and cover its fixed costs. Find the break-even points for this function. Page 10

11 MATH 1306 Handout # 1 Answers To Odd-numbered Problems 1a. i) C "! ii) B # 1b. i) C "& ii) B &ßB$ 1c. i) C "" ii) none 1d. i) C! ii) B %ßB! 1e. i) C! ii) B #ßB!ßB# 1f. i) C! ii) B!ßB" 1g. i) C ' ii) B "ßB"ßB$ 3a. Verbal description: All real numbers between # and #Þ&, inclusive Interval notation: Ò #ß #Þ&Ó 3b. B #ßB"ßB# 3c. C% 5a. Verbal description: All real numbers between *Þ& and "', inclusive Interval notation: Ò*Þ&ß "'Ó The daily hours of operation for the NASDAQ are 9:30 am to 4 pm. 5b. No 5c. MÐ"'Ñ stands for the closing value of the NASDAQ (at 4 pm) on a recent trading day. 5d. MÐ"%Ñ ")%' 5e. MÐ*Þ&Ñ ")&! 5f. >ß M 5g. hours, points Page 11

12 7a. Verbal description: All real numbers between ' and #!, inclusive Interval notation: Ò'ß #!Ó 7b. GÐ"$Ñ stands for the number of cars in thousands traveling on the Expressway at 1 pm. 7c. GÐ#!Ñ ' 7d. GÐ)Ñ "! 7e. >ß G 7f. hours, thousands of cars 9a. EÐ!Ñ #!!; The tank initially has #!! gallons of water. 9b. ; After minutes, the tank has been completely drained - it contains EÐ#!Ñ! #! no water. Page 12