Deck 1: A Library of Functions

Question 1

A school library opened in 1980.In January, 2000 they had 25,000 books.One year later, they had 25,420 books.Assuming they acquire the same number of books at the start of each month, how many books did they have in January, 2003?

Answer 1

26,260

Question 2

The following table defines three functions for $0 \leq x \leq 4$
.The function $y_{1}$
is most likely
$$\begin{array}{l}x\\\begin{array} { l l l } &&y _ { 1 } && y _ { 2 } && y _ { 3 }\end{array}\\\begin{array} { c c c c } 0 & 4.65 & 4.65 & 4.65 \\1 & 7.44 & 5.11 & 3.79 \\2 & 11.90 & 5.97 & 2.93 \\3 & 19.05 & 9.55 & 2.07 \\4 & 30.47 & 15.28 & 1.21\end{array}\end{array}$$

A)exponential
B)neither
C)linear

Answer 2

exponential

Question 3

On the interval between E and F, the function graphed below is: Mark all that apply.

A)increasing
B)decreasing
C)concave up
D)concave down

Answer 3

decreasing
concave up

Question 4

A school library opened in January of 1980.In January, 2000 they had 35,000 books.One year later, they had 35,490 books.Assuming they acquire the same number of books at the start of each month, how many books did they have in July of 1980?

Question 5

Suppose the Long Island Railroad train from East Hampton to Manhattan leaves at 4:30 pm and takes two hours to reach Manhattan.It waits two hours at the station and then returns, arriving back in East Hampton at 10:30 pm.Draw a graph representing the distance of the train from the Farmingdale station in East Hampton as a function of time from 4:30 pm to 10:30 pm.The distance from East Hampton to Manhattan is 150 miles.

Question 6

Draw a graph which accurately represents the temperature of the contents of a cup left overnight in a room.Assume the room is at 70° and the cup is originally filled with water slightly above the freezing point.

Question 7

A pond has a population of 500 frogs.Over a ten-year period of time the number of frogs drops quickly by 20%, then increases slowly for 5 years before dropping to almost zero.Does the following graph accurately represent the number of frogs in the pond over the ten-year period of time?

Question 8

The empirical function W = f(t), given in the graph below, comes from the Wall Street Journal, September 4, 1992.From the graph, determine the domain of the function.

A)September 1989 to August 1992
B)2810 to 4090
C)0 to 4200
D)January 1988 to January 1993

Question 9

A bar of soap starts out at 100 grams.What is the formula for the quantity S grams of soap remaining after t days if the decrease is 15% per day?

Question 10

Suppose we buy quantities and , respectively, of two goods.The following graph shows the budget constraint , where and are the prices of the two goods and k is the available budget.If the budget is doubled, but prices remain the same, what is the intercept of the new budget constraint?

A)
B)
C)
D)

Question 11

A school library opened in 1980.In January, 2000 they had 40,000 books.One year later, they had 40,410 books.Assume that they acquire the same number of books at the start of each month.If you graph the function with domain 1980-2010, describe the y-intercept of the graph in the context of the problem.

A)The number of books the library had in 1980
B)The number of books the library will have in 2010
C)The year the library had 40,000 books
D)The year the library had no books

Question 12

Suppose a 40 ${ } ^ { \circ } F$ container of water is placed in the freezer overnight.The next morning, it is put on the counter in a 70 ${ } ^ { \circ } F$ room and then at the end of the day heated to the boiling point.What is the range of the function?

A)From 40 ${ } ^ { \circ } F$ to 70
${ } ^ { \circ } F$ .
B)From the middle of the night until the middle of the next day.
C)From the first evening until the end of the next day.
D)From 32 degrees F to 212 degrees F.

Question 13

Write a formula representing the function that says: The circumference of a circle is proportional to the diameter of the circle.

Question 14

Suppose we buy quantities and , respectively, of two goods.The following graph shows the budget constraint , where and are the prices of the two goods and k is the available budget.If the price of the first good is doubled, but the other values are unchanged, what is the intercept of the new budget constraint?

A)
B)
C)
D)

Question 15

The illumination, I, of a candle is inversely proportional to the square of its distance, d, from the object it illuminates.Write a formula that expresses this relationship.

Question 16

A bar of soap starts out at 175 grams.What is the formula for the quantity S grams of soap remaining after t days if the decrease is 10 grams per day?

Question 17

If , find and simplify the difference quotient .

A)1
B)
C)
D)

Question 18

Harley Davidson (ticker symbol HOG)stock prices dropped sharply in late 2008.Series 1 in the graph below shows the actual prices at the end of each week.The trend over time is approximately linear; and the graph of a possible linear model is given by Series 2.Based on the data given, find the linear model and use it to approximate the stock's price on November 30, 2008, assuming the current trend continued.

Question 19

A function is linear for x $\le$ 2 and also linear for x $\ge$ 2.This function has the following values: f(-4)= 8; f(2)= 5; f(4)= 11.Find formula(s)(or equation(s))which describe this function.

A) $f(x)=\left\{\begin{array}{ll}\frac{1}{2} x+4, & x \leq 2 \\-3 x+11, & x \geq 2\end{array}\right.$
B) $f(x)=\left\{\begin{array}{cc}2 x+6, & x \leq 2 \\\frac{1}{3} x-11, & x \geq 2\end{array}\right.$
C) $f(x)=\left\{\begin{array}{cc}-2 x+9, & x \leq 2 \\3 x-1, & x \geq 2\end{array}\right.$
D) $f(x)=\left\{\begin{array}{cl}-\frac{1}{2} x+6, & x \leq 2 \\3 x-1, & x \geq 2\end{array}\right.$

Question 20

The curve W = f(t), given in the graph below, comes from the Wall Street Journal, September 4, 1992.W is a function.

Question 21

Given the function , give a formula for the inverse function of f(x).

Question 22

Is the function graphed in the following figure invertible?

Question 23

The following functions represent exponential growth or exponential decay.Mark the one(s)that represent exponential growth.

A)
B)
C)
D)

Question 24

Lisinopril is an ACE inhibitor derived from the venom of a Brazilian pit viper frequently used in the treatment of hypertension.Because of Lisinopril's relatively long half life of 12 hours, patients need to take a dose just once per day.A patient takes his first dose, 20 mg, at 6 pm on Saturday.
a)How many hours does it take for the amount of Lisinopril in the patient's body to decrease to 16 mg? Round to two decimal places.
b)How many milligrams remain in the patient's body right before he takes his next 20 mg at 6 pm on Sunday? Round to two decimal places

Question 25

Given the information in the table and the following conditions, find .
• f(x)is even.
• g(x)is odd.
• h(x)=

Question 26

Given the function , find and simplify .

Question 27

Is the function odd, even , or neither?

Question 28

The cost of shipping r kilograms of material is given by the function C = f(r)= 175+ 6r.Find a formula for the inverse function.

Question 29

Write an equation for the graph obtained by shifting the graph of vertically upward by 3 units, followed by vertically stretching the graph by a factor of 2 and reflecting the graph across the axis.

Question 30

Given the graphs of and in the following figure, estimate g(f(5)).

Question 31

Joe invested $20,000 in the stock market, and Sam invested $20,000.Joe's investment increased in value by 5% per year for 10 years.Sam's investment decreased in value by 10% for 5 years and then increased by 10% for the next 5 years.At the end of the 10 years, whose investment was worth more, Joe's or Sam's?

Question 32

A bakery has 300 lbs of flour.If they use 10% of the available flour each day, how much do they have after 20 days? Round to the nearest pound.

Question 33

Given the function find and such that

A) and
B) and
C) and
D) and

Question 34

If the graph of y = f(x)is shrunk vertically by a factor of 1/2, then shifted vertically by 4 units, then stretched vertically by a factor of 2, the resulting graph the same as the original graph.

Question 35

The graph of y = f(x)is shown in the first figure.What graph is shown in the second figure?

A)
B)
C)
D)

Question 36

Given the function , which of the following is equivalent to ?

A)
B)
C)
D)

Question 37

Given the function , which of the following is equivalent to ?

A)
B)
C)
D)

Question 38

One of the graphs below shows the rate of flow, R, of blood from the heart in a man who bicycles for twenty minutes, starting at t = 0 minutes.The other graph shows the pressure, p, in the artery leading to a man's lungs as a function of the rate of flow of blood from the heart.Estimate .

A)26.5
B)23
C)17.5
D)13

Question 39

In the book One Grain of Rice, a girl receives a reward that starts with one grain of rice on day one, two grains on day two, four on day three and eight on day four.Each day, she receives double the number of grains of rice.How many grains of rice does she receive on the 30th day?

A)900
B)1,073,741,824
C)4,640,650,289
D)60

Question 40

Give a possible formula for the function in the following figure:

Question 41

The following describes a function that must be invertible: N(p)is the number of words on page p of a science fiction novel.

Question 42

In 1909, the Danish biochemist Sören Peter Lauritz Sörensen (1868-1939)introduced the pH function as a measure of the acidity of a chemical substance: pH = f([H+])=

Question 43

If the size of a bacteria colony doubles in 8 hours, how many hours will it take for the number of bacteria to be 11 times the original amount? Round to 2 decimal places.

Question 44

Find the equation for the line L shown below.

A)
B)
C)
D)

Question 45

Suppose f(x)and g(x)are both increasing functions.
a)Must f(g(x))be an increasing function? If so, explain why.If not, give an example.
b)Must f(g(x))be invertible? If so explain why.If not, give an example.

Question 46

Cramped Quarters Island is a tiny island which, although able to support a limited population of rabbits, doesn't have enough space or food supplies to support unlimited exponential growth.It is suggested that if Q(t)= population of rabbits on Cramped Quarters Island at time t, then the quantity (800 - Q(t))will be an exponentially decaying function of t.Given that there were 400 rabbits at time t = 0, and 500 rabbits one year later, find the general formula for Q(t), the population of rabbits on Cramped Quarters Island at time t and use it to determine the rabbit population after 3 years (to the nearest rabbit).

Question 47

Suppose there is an initial population of 100 rabbits on Prosperity Island.Assuming that the rabbits have more than enough of everything they need to live prosperously, we might expect the population to grow exponentially.If so, find a formula for P(t), the number of rabbits on Prosperity Island at time t, given that after one year there are 140 rabbits on the island.(Assume t is in years.)Use this formula to determine how many years it will be before there are 400 rabbits on Prosperity Island.Round to 1 decimal place.

Question 48

Cramped Quarters Island is a tiny island which, although able to support a limited population of rabbits, doesn't have enough space or food supplies to support unlimited exponential growth.It is suggested that if Q(t)= population of rabbits on Cramped Quarters Island at time t, then the quantity (800 - Q(t))will be an exponentially decaying function of t.If at time t = 0 there were 250 rabbits, and the population was increasing at an instantaneous rate of 100 rabbits per year, give the general formula for .

Question 49

The graph of a function y = f(x)is given below.Sketch a graph of it's inverse function on the same axes.

Question 50

Solve for x.Round to 2 decimal places.

Question 51

Solve for x.Round to 3 decimal places.

Question 52

The following describes a function that must be invertible: A woman takes a tablet of 10 mg of a common allergy medicine with a half life of approximately 8 hours.L(t)gives the amount of medicine in her system at time t where 0 < t < 24.

Question 53

A one-to-one function is described in words as follows.Take x and multiply by 4.Then add 3 and cube the result.Describe the inverse function in words.

Question 54

What is the doubling time in years of prices which are increasing by 8% a year? Round to two decimal places.

Question 55

An exponentially decaying substance was weighed every hour and the results are given below.What is the approximate half-life, in hours, of the substance? Round to 1 decimal place.

Question 56

Suppose that N(t)= 100,000,000 · gives the population of a certain country t years after a census was taken.A historian has a collection of documents that are not dated, but do refer to the population of this country at several times.In order to help the historian date these documents, find the inverse function for the function N.Round to whole numbers.

Question 57

In 1992, the Population Crisis Committee wrote:
Large cities in developing countries are growing much faster than cities in the industrialized world ever have.London, which in 1810 became the first industrial city to top 1 million, now has a population of 11 million.By contrast, Mexico City's population stood at only a million just 50 years ago and now is 20 million.
Assume that the instantaneous percentage growth rates of London and Mexico City were constant over the last two centuries.How many times greater is Mexico City's percentage growth rate than London's (to one decimal place)?

Question 58

The number of bacteria in milk grows at a rate of 10% per day once the milk has been bottled.When the milk is put in the bottles, it has an average bacteria count of 500 million per bottle.Suppose milk cannot be safely consumed if the bacteria count is greater than 2.9 billion per bottle.How many days will the milk be safe to drink once it has been bottled? Round to the nearest integer.

Question 59

Here are some data from a Scientific American article on Old World monkeys.From the data presented, give an approximate formula for C = cranial capacity (in cm³)as a function of A = arc length of skull (in cm).Round to one decimal place. Cranial Capacity of contemporary Old World monkeys is related to arc length of skull as shown.

Question 60

The following describes a function that must be invertible: R(t)is the revenue earned by a shoe shop on the $t ^ { t h }$ th day of the year.

Question 61

Graphically find a solution to the equation in the interval .Give your answer accurate to 2 decimal places.

Question 62

One of the functions below is a quadratic, one is a cubic, and one is a periodic function.Which one is periodic? $$\begin{array}{l}\begin{array} { r | r } \hline x & f ( x ) \\\hline 0.2 & - 0.42 \\0.4 & - 0.65 \\0.6 & 0.96 \\0.8 & - 0.15 \\1.2 & 0.84 \\\hline\end{array}\\\\\begin{array} { r | r } \hline x & g ( x ) \\\hline 1.3 & 0.41 \\1.7 & 0.81 \\2.5 & 0.65 \\3.0 & - 0.10 \\3.5 & - 1.35 \\\hline\end{array}\\\\\begin{array} { r | r } \hline x & h ( x ) \\\hline 0.5 & - 1.13 \\1.2 & 0.13 \\1.8 & 0.03 \\2.0 & 0.00 \\2.2 & 0.05 \\\hline\end{array}\end{array}$$

A)g(x)
B)f(x)
C)h(x)

Question 63

Consider the functions and .Which has a larger vertical intercept?

A)f(x)
B)g(x)

Question 64

Using the standard viewing rectangle (-10 $\le$ x $\le$ 10, - 10 $\le$ y $\le$ 10), I graphed a cubic polynomial and saw two vertical lines, as shown.There must be another root outside of the window. x

Question 65

Temperatures in Town A oscillate daily between 30° F at 4am and 60° F at 4pm.The temperature in Town B is consistently 10° F colder than in Town A.What does the formula represent?

A)Temperature in Town A, in terms of time where time is measured in hours from 4am.
B)Temperature in Town B, in terms of time where time is measured in hours from 4am.
C)Temperature in Town A, in terms of time where time is measured in hours from 4pm.
D)Temperature in Town B, in terms of time where time is measured in hours from 4pm.

Question 66

Tornados are classified in several ways.A tornado's classification on the Fujita Scale as F1 through F5 is most commonly cited.Another classification of tornados is by path length, given by the formula where L is the length of the tornado's path in miles.The Binger, Oklahoma tornado of 1981 was an F4 (originally thought to be an F5)whose path was 16 miles in length.What was its Pl classification?

A)P1
B)P2
C)P3
D)P4
E)P5

Question 67

Delia runs the 3.1 miles from home to the park at 6 mph, jumps on her bike and returns home in 12 minutes.
a)Sketch a well-labeled graph of Delia's distance from home as a function of time.
b)Find the slope of each segment of the graph and interpret their meaning.
c)What does it mean to say that Delia's velocity is inversely proportional to the time she takes for her run/ride?

Question 68

Sketch a well-labeled graph of a periodic function such that :
-- f(0)= 850
-- the period is 12
-- the amplitude is 550.
Then write a few sentences illustrating how such a function might apply to a scorpion population.

Question 69

Write in factored form the equation of the polynomial graphed below.All key features are shown.

Question 70

Make a graphical sketch of the following function.Include a scale on the x- and y-axis.

Question 71

Using base e and transformations, find a formula for the exponential function shown in the graph below.

Question 72

The solid curve below is a portion of the graph of and the dashed curve is a portion of the graph of .The domain of both functions is all real numbers.Which of the following statements are true.Check all that apply.

A)At x = 1, .
B)For all x > 1, .
C) .
D)There is only one value of x for which .

Question 73

What is the period of the function ?

A)
B)
C)
D)The function is not periodic

Question 74

Coulomb's law says that the electrical force between two charged objects is directly proportional to the product of the quantity of charge on the objects and inversely proportional to the square of the distance between the objects.Let and be the charge on the two objects.Let d be the distance between the objects and F be the electrical force between them.Translate Coulomb's Law into mathematical symbols.

A)
B)
C)
D)

Question 75

In nature, the population of two animals, one of which preys on the other (such as foxes and rabbits)are observed to oscillate with time, and are found to be well approximated by trigonometric functions.The population of foxes is given by the graph below.What is the amplitude?

Question 76

Give a formula for the following sinusoidal function as a transformation of

Question 77

A spherical cell takes in nutrients through its cell wall at a rate proportional to the area of the cell wall.The rate at which the cell uses nutrients is proportional to its volume.Which of the following expresses the rate at which the cell uses nutrients as a function of its radius, r.(A is a constant.)

A)
B)
C)
D)

Question 78

At the point , the function has

A)A local maximum
B)A local minimum
C)Neither

Question 79

Find a possible equation for the following function given a root at (0,0), a vertical asymptote at y=-1 and horizontal asymptote at x=2.

Question 80

What is the period of the function