Functions Questions and Answers

Question 2 Graphs of Functions 50 points total The below graph is a function Using this graph answer the below questions 4 3 2 1 0 1 2 3 5 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 Question 2 1 Part A 17 points Define the open intervals where the function was increasing Put a comma between each interval if needed Question 2 2 Part B 17 points Define the open intervals where the function was decreasing Put a comma between each interval if neede Question 2 3 Part C 16 points

Alyssa and Gabriel play the following game Alyssa rolls a die If she rolls a 1 2 or 3 Gabriel gives Alyssa 5 If Alyssa rolls a 4 or 5 Gabriel gives Alyssa 2 However if Alyssa rolls a 6 she gives Gabriel 11 a Determine Alyssa s expectation b Determine Gabriel s expectation a Alyssa s expectation is dollars Round to the nearest cent

A rack of 15 billiard balls is shown If one ball is selected at random determine the odds against it containing a number greater than or equal to 9 The odds are Simplify your answer 15

By cutting away an x by x square from each corner of a rectangular piece of cardboard and folding up the resulting flaps a box with no top can be constructed Volume as a function of x 4x 136x 1056x Domain of the function for volume If the piece of cardboard is 44 inches long by 24 inches wide find a function in the variable x giving the volume of the resulting box T8L 0 12 U 22 00 FRL X TIL Determine the domain of the function for volume Enter your answer using interval notation THL MY NOT

Limits Continuity Course Packet on limits at infinity of rational functions The Penn State Math Department s total costs in dollars for producing x units of their award winning calculus tutorial videos 72x3 9x3 What is the limiting value of the total cost when the number of units produced grows infinitely large dollars C x 300 1 26x x 61x 1 82

In a game of craps you roll two fair six sided dice The sides of each die are labeled 1 through 6 Whether you win or lose depends on the sum of the numbers occurring on the tops of the dice Let x be the random variable that represents the sum of the numbers on the tops of the dice a What values can x take on Enter your answers as a comma separated list b What is the probability distribution of these x values Enter the probabilities in ascending order of the x values Round your answers to three decimal places Probabilities 0 028 0 056 0 083 0 111 0 139 0 167 0 139 0 111 0 083 0 056 0 028

011 Limits Continuity Course Packet on limits at infinity of rational functions 29x5 68x6 75x7 96x 0 34 If you need to use or enter INFINITY or INFINITY respectively Evaluate lim X 7 8

Consider the following a Draw a tree diagram to display all the possible outcomes that can occur when you flip a coin and then toss a six sided die sasta b How many outcomes contain a head and a number greater than 2 G c Assuming the outcomes displayed in the tree diagram are all equally likely what is the probability that you will get a head and a number greater than 2 when you flip a coin and toss a six sided die Round your answer to three decimal places

Consider the following equation of a line 2x 5y 15 a Find the slope and y intercept if possible of the equation of the line algebraically Use a graphing utility to verify your answer If an answer is undefined enter UNDEFINED Enter NONE if there is no y intercept slope y intercept x y b Sketch the line by hand Use a graphing utility to verify your answer No Solution 10 9 8 7 6 5 4 3 2 1 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 8 9 1 2 3 4 5 6 7 8 9 10 Fill Graph Layers After you add an object to the graph you can use Graph Layers to view and edit its properties

A person created a large pond in their backyard and they wanted to stock it with fish The first year they started with 8 fish and then watched the growth of the population of the fish over the years They found the population could be modeled by 67 1 19 19 341 Where y is the population size of fish and x is the number of years after they began What do you think will be the best estimate for the maximum number of fish the pond will sustain based on their model a 86 b 8 c 19

Not yet answered Marked out of 1 00 The coefficient of correlation O a is the square root of the coefficient of determination O b is the same as R O d can never be negative is the square of the coefficient of determination

If the coefficient of correlation is 0 60 then the coefficient of determination is a 0 36 O b 0 6 OC 0 77 O d 0 36

A long year end status report for work is 81 pages long You need to print 19 copies for a meeting next week How much is the paper going to cost for those reports Paper is sold in reams 500 pages for 3 99 each Give your answer to the nearest cent only for the paper you yon

Find the value of the expression See Examples 3 5 Objective 2 2 5 4

Show that B is the inverse of A AB BA 3 1 B 52 A 2 1 5 3 I I

6 Give the equation of the function described below and then sketch its graph The graph of y x is shifted 4 units to the left Then the graph is shifted 3 units downward Equation 7 y f x 1 f x x W

6 Give the equation of the function described below and then sketch its graph The graph of y x is shifted 4 units to the left Then the graph is shifted 3 units downward Equation y axis

Which of the following statements are true More than one answer may be possible Select all that appl o a c 0 O o a s o o a a H 0 O o a 2 o

9 Prove that a third degree polynomial function always has exactly one Point of Inflection T2

Solve the compound inequality and write the answer using interval notation Simplify your answer completely x 1 9 or x 1 9

Use the revenue and cost graphs on the right to find the approximate cost of manufacturing 28 000 cell phones What is the approximate cost of manufacturing 28 000 phones Dollars in millions A 5 4 3 2 1 0 Cost

Determine the gain of the amplifier in Fig P4 14 R1 3 6 k0 R2 25 kn V in Figure P4 14 Notes on entering solution R R Enter your solution as a constant Give your answers with a positive or hundredth ex 5 is entered as 5 00 negative sign Enter your solution to the nearest OV

b Find the values of the constants A and B so that the following function is continuous every where Ax B if x 1 x 2x 3 Ax B if 1 x 2 4 if x 2

4 A function y f x is defined by the graph below Answer the following questions Approximate when necessary Assume the graph is expanded indefinitely 5 4 3 2 2 0 21 31 f List all zeros of the function 4 5 a Find the domain b Find the range c What is f 4 d What is f 3 e Estimate the x value s for which f x 2

logists found the population of two invasive fish species in a large lake was changing exponentially after the lake was first surveyed Here are graphs showing the growth of each fish population after the survey began in years Population A Population 45000 aood 35000 30000 25000 20000 15000 20000 1 time years 2 A Which fish population was greater when the biologist first surveyed the lakes D Which population has a Asymptote of y 1500 3 Population A was greater than population B 5 Population B 48000 45000 42000 39000 Population A Population Population B 36000 33000 30000 27000 24000 B What does the point 3 27000 of Population B represent in this situation Describe the point in the form of function notation In years the f x population was 27 000 21000 18000 15000 12000 9000 6000 3000 E Describe the domain of each function What does the domain mean in this situation What are the restrictions of the domain Population 13 Since the point is 0 1500 that arise from the context 2 time years 1 3 C What is the growth factor for each fish population Population A Population B F Describe the range of each function What does the range mean in context of the situation What are the restrictions of the range that arise from the context Population A Population B

Use the graph to help you answer the questions that follow You will not be graded on any adjustments made to the graph Tool tip Click and drag one or both of the curves Curves will snap into position so if you try to move a curve and it snaps back to its original position just try again and drag it a little farther J TUITION Thousands of dollars 15 Admission to New York University Supply bemand 12 18 NUMBER OF STUDENTS Thousand 20 The equilibrium level of tuition at NYU is 24 Demand 101 Supply C per academic year If NYU sets its tuition at this price the number of student applications the number of openings available Now suppose that the tuition for NYU is set at 30 000 by New York s state assembly At this level of tuition the number of student applications the number of openings available wit Suppose that in its latest issue a popular magazine publishes information about colleges in the United States The magazine declares NYU to be America s best college Adjust the previous graph to show the effect this will have on the market for admission to NYU The new equilibrium level of tuition at NYU is per academic year it the majapine declares NrU to be America s best college and the tuition for Nu is set at 30 000 by New York s assembly NYU will smission than there are coenings C

Again the following graph shows the supply of and demand for Sam s athletic services Suppose that the NCAA revokes its rule so universities ca pay student athletes any wage with no limitations Use the black point to indicate the wage the University of California Berkeley will pay Sam under these conditions Then shade Berkeley s consume surplus using the green point triangle symbol Finally shade Sam s total wage revenue using the purple point diamond symbol 23X22 WAGE Thaad Demand Sam s Athletic Services Supply Equilibrium Wage A Total Wo

Adjust the following graph to show the effect of the introduction of the health insurance plan on the market of x rays Tool tip Click and drag one or both of the curves Curves will snap into position so if you try to move the curve and it snaps back to its origin position just try again and drag it a little farther PRICE OF X RAYS Market for X Rays Supply Demand O Demand Supply

Let x be the average number of employees in a group health insurance plan and let y be the average administrative cost as a percentage of claims O x y a Make a scatter diagram of the data and visualize the line you think best fits the data 40 40 y average cost as percentage y average cost as percentage r 37 7 15 34 74 40 37 34 28 17 35 30 25 20 USE SALT 35 30 25 20 10 20 30 40 50 60 70 x average number of employees 10 20 30 40 50 60 70 x average number of employees O y average cost as percentage DO y average cost as percentage 35 30 25 20 40 35 30 25 20 10 20 30 40 50 60 70 x average number of employees 10 b Would you say the correlation is low moderate or strong positive or negative Olow and negative O moderate and positive O strong and negative Olow and positive O moderate and negative Ostrong and positive 20 30 40 50 60 70 x average number of employees Q c Use a calculator to verify that Ex 133 Ex 6 915 Ey 156 Ey2 5 198 and Exy 3 099 Computer Round your answer to four decimal places As x increases does the value of r imply that y should tend to increase or decrease Explain Given our value of r y should tend to remain constant as x increases Given our value of r we cannot draw any conclusions for the behavior of y as x increases CiVeR chould tond

Determine the ordered pair for the point labeled A in the figure below x 5 5 4 3 2 1 4 3 2 1 1 2 3 T 1 2 3 4 5 X

Compute the slope of the line If an answer is undefined enter UNDEFINED 0 121 74 and 5 176 8 b Classify the line containing the points as tending upward tending downward horizontal or vertical tending downward tending upward horizontal vertical

15 20 15 10 0 5 15 Determine if x and y have a positive association a negative association or neither positive negative neither 10 X 20

A nutritionist surveys seven patients and records the typical number of minutes they exercise per day and their body mass index BMI The results are shown in the following table BMI Construct a scatterplot of the data in which the horizontal axis corresponds to minutes of exercise and the values start at 0 and end at 60 and the vertical axis corresponds to BMI and the values start at 0 and end at 30 BMI 30 25 20 15 10 5 30 25 20 15 10 Minutes of Exercise 60 30 50 0 15 40 10 5 BMI 15 20 14 30 26 18 29 5 10 15 20 25 30 35 40 45 50 55 60 Minutes of Exercise 5 10 15 20 25 30 35 40 45 50 55 60 Minutes of Exercise BMI BMI 30 25 20 15 10 12 5 30 25 20 15 10 5 5 10 15 20 25 30 35 40 45 50 55 60 Minutes of Exercise 5 10 15 20 25 30 35 40 45 50 55 60 Minutes of Exercise

A scatterplot is given 20 15 10 10 15 20 Determine if x and y have a positive association a negative association or neither positive negative neither

If f x 31 3 x 11 and g x 2x2 20 compute the composition fog 5 fog 5 35 ns X

Let f x f 5 4 5 f 5 h X Simplify each of the following expressions 4 5 h f 5 h f 5 f 5 h f 5 4h 5 5 h 1 5 h X Make sure to combine terms by getting a common denominator Make sure to cancel out the factor of h in the denominator

Functions Course Packet on the composition of functions If f x 11 and g x 26x 10 compute the compositions fog 1 and gof 1 5x 6 fog 1 gof 1

Let f x 5x 7x 2 and g x 3x 5 Determine the rule for the composite functions f o g and go f fog x NS gof x

First find the low and high numbers of the five number summary for the given data set Recall that the five number summary is used to create a box and whisker plot The five number summary consists of the lowest value the first quartile Q the median the third quartile Q3 and the highest value of a data set The given data set 7 10 10 11 12 12 13 14 15 is ordered We see that the lowest value is 7 7 and the highest value is 15 3 Step 2 Now find the median Recall that the median is the central value of an ordered distribution Note that the median is also the second quartile Q odd number of data values as we have here the median is the middle data value of the ordered For an odd data set The median of the ordered data set 7 10 10 11 12 12 13 14 15 is 12 Step 3 Next find Q Recall that the first quartile Q is the median of the lower half of the data That is the median of the data located below the Q position 7 10 10 11 12 12 13 14 15 There are nine values in the whole data set but there are only median The first quartile Q will be equal to the median of these values 15 Consider the ordered list of data values The median Q which was determined in the last step has been underlined for clarity sum of middle two values 2 The middle two values of the lower half of the data set are 10 and Now calculate the first quartile Q 10 10 2 median 12 2 Since the lower half of the data has an even number of data values we will now use the rule to find the median of an even number of data values x values to the left of the underlined

A survey was done at Center Hospital to determine how long in months clerical staff had been in their current positions The responses in months of 20 clerical staff members were as follows 2 6 7 13 14 16 a Select the box and whisker plot 0 10 20 30 40 50 60 70 80 17 21 23 23 23 0 10 20 30 40 50 60 70 80 Find the interquartile range IQR 10 40 26 50 The medians are about equal for clerical staff and narrower for clerical staff compared to that of 26 27 30 b Below is the box and whisker plot of data from a survey of 20 nurses at Center Hospital 31 32 32 41 73 10 20 30 40 50 60 70 80 20 30 Compare this plot with the one in part a Discuss the locations of the medians the locations of the middle halves of the data banks and the distances from Q and Q3 to the extreme values 10 20 30 40 50 60 70 80 nurses The interquartile range is nurses Q is farther from C the minimum for clerical staff compared to

The positions of town A town B town C and town D are marked on a Cartesian plane Given 2y 4x 12 represent the equation for the straight line connecting town A and town C while y 4x represent the equation for the straight line connecting town B and town D If town M is the intesection of both equations state the distance town M from x axis

Log AB B Log A If 0 5 0 01 what is the value of n

6 A piece of silver wire has a diameter of 0 510 mm If silver has a density of 10 5 g cc how long in meters should you cut a piece of wire to obtain 0 0178 moles of silver

ASK YOUR TEACHER A country s census lists the population of the country as 258 million in 1990 281 million in 2000 and 298 million in 2010 Fit a second degree polynomial passing through these three points Let the year 2000 be x 0 the year 2020 be x 20 and let p x represent the population in millions p x million Use this polynomial to predict the populations in 2020 and in 2030 2020 million 2030 million PRACTICE ANOTHER

Round 0 00026546 to six decimal places

Classify 16 4 x expression O equation Identify the variables coefficients and constants Enter your answers as a comma separated list If there is no answer enter NOR variables coefficients as an algebraic expression or an equation portante

Four different linear functions are represented below 10 6 3 Function 1 10 Function 3 y x 4 Answer the following questions 6 8 10 Function 2 X 2 1 0 1 2 y 4 st 1 2 5 8 Function 4 The slope is 5 and the y intercept is 1 a Which function has the graph with a y intercept closest to 0 O Function 1 Function 2 Function 3 Function 4 b Which functions have graphs with y intercepts greater than 2 Check all that apply Function 1 Function 2 Function 3 Function 4 c Which function has the graph with the greatest slope Function 1 Function 2 Function 3 Function 4

Four different linear functions are represented below 10 8 Function 1 10 11 aft Function 3 y x 2 answer the following questions 6 8 10 Function 2 a Which function s graph is the least steep X 2 1 0 1 2 y 11 6 1 4 9 Function 4 The slope is 2 and the y intercept is 4 Function 1 O Function 2 Function 3 Function 4 b Which functions have graphs with slopes less than 4 Check all that apply Function 1 Function 2 Function 3 Function 4 c Which function has the graph with the greatest y intercept O Function 1 Function 2 Function 3 Function 4

For the linear programming problem Maximize f 6x1 7x2 subject to 50 60 6x19x2 3x1 2x2 X1 0 X2 0 A Fill in the initial tableau feel free to just draw on paper x1 x2 S t

1000 900 800 a range 700 600 Amount 500 of flour grams 400 300 100 Time minutes Write your answers as inequalities using x or y as appropriate Or you may instead click on Empty set or All reals as the answ h demain 0 0 0 0 Oso 8 08 20 Empty set a reals