ConcepTests For Math 112

Prepared by Richard Cangelosi

and Mariamma Varghese

2

If the sum of two numbers is 14, and their difference is 2, what is the product of the two numbers?

A. 24

B. 28

C. 40

D. 45

E. 48

F. None of these.

3

Compare the quantity in Column A with the quantity in Column B given the condition:

Column A Column B

2

200

x yxy

x y

(A)The quantity in Column A is greater

(B) The quantity on Column B is greater

(C) The two quantities are equal

(D)The relationship cannot be determined from the information given

4

Which of the following equals the reciprocal of

where ?

(A) .

(B) .

(C) .

(D) .

(E) .

1x y 01x y

1 yx

yx

1y

x

1x

xy

1y

xy

5

Which of the following could be a graphical representation of

6 52 53

xx A.

B.

C.

D.

E.

0

0

0

0

0

6

Compare the quantity in Column A with the quantity in Column B given the following condition.

Column A Column B

3 3 6x

x 0

(A)The quantity in Column A is greater

(B) The quantity on Column B is greater

(C) The two quantities are equal

(D)The relationship cannot be determined from the information given

7

Compare the quantity in Column A with the quantity in Column B given the condition:

Column A Column B

1x x

x 0

(A)The quantity in Column A is greater

(B) The quantity on Column B is greater

(C) The two quantities are equal

(D)The relationship cannot be determined from the information given

8

Compare the quantity in Column A with the quantity in Column B given the condition:

Column A Column B0 r t

rt

tr

(A)The quantity in Column A is greater

(B) The quantity on Column B is greater

(C) The two quantities are equal

(D)The relationship cannot be determined from the information given

9

Solve for x given2/3

2 4 25x

(A) -5.8, 5.8

(B) -4.6, 4.6

(C) 21

(D) -11, 11

(E) None of these

10

Solve for x given: 2 5 5x x

(A) 1/5

(B) -2/5

(C) 1/5, -2/5

(D) 1/10

(E) None of these

11

Determine the domain of 1

4 2 5x

(A) .

(B) .

(C) .

(D) .

(E) None of these

2,5

5,2

5,2

,25

12

Below is part of the graph of . What is the average rate of change of f between ?

y f x0 x b

(A) .

(B) .

(C) .

(D) .

(E) . None of these.

( )f b f ab a

ab

ab

ba

13

The graph to the right gives a complete representation of the function, . For what values of x is ?

y f x 0f x

(A) . -3, -2, -1

(B) . -3, -2, -1, 2, 3

(C) .

(D) .

(E) .

(F) . None of these.

3 1 and 2 3x x

3 1 and 2 3x x

0x

14

Determine which of the following graphs are even functions.

I II III

(A) I only

(B) II only

(C) III only

(D) None of these

15

Compare the quantity in Column A with the quantity in Column B given the condition:

Column A Column B

(A)The quantity in Column A is greater

(B) The quantity on Column B is greater

(C) The two quantities are equal

(D)The relationship cannot be determined from the information given

The perimeter of a rectangular desktop is 10 feet. Its length is 1 foot greater than its width.

The area of the desktop 5 sq. ft.

16

In the figure below, ABCD and QRST are rectangles where QR= ½ AB and RS = ½ BC. What fraction of the region bounded by ABCD is NOT shaded?

C

A

B

T

S

R

Q

D

(A) (B) (C) (D) (E)14

13

12

23

34

17

A machine can stamp 20 envelops in 4 minutes. How many of these machines, working simultaneously, are needed to stamp 60 envelops per minute?

(A) 5

(B) 10

(C) 12

(D) 20

(E) 24

18

Draw a graph to represent the following situation. Label axes and intercepts.

A boy who is 4 inches tall is growing at a rate of 2 inches per year.

19

Draw a graph to represent the following situation. Label axes and intercepts.

A person pays $10 per week to a friend in order to pay off a $100 loan.

20

It takes Carla three hours to drive to her brother’s house at an average speed of 50 miles per hour. If she takes the same route home, but average speed is 60 miles per hour, how long does it take her to get home?

(A)2 hours

(B) 2 hours and 14 minutes

(C) 2 hours and 30 minutes

(D)2 hours and 45 minutes

(E) 3 hours

21

Which of the following operations will affect the zeros of ? Assume . y f x

(i) (ii) (iii) (iv)

(A) (i) and (ii) only

(B) (ii) and (iv) only

(C) (i), (ii) and (iii) only

(D) (ii), (iii) and (iv) only

(E) All of them

(F) None of them

Kf x f x K f x K f Kx

0K

22

If y = f(x) is an even function, then which of the following functions are also even?

(i) Kf(x) (ii) f(x + K) (iii) f(x) + K

(A) (i) only

(B) (ii) only

(C) (ii) and (iii) only

(D) (i) and (ii) only

(E) (i) and (iii) only

(F) All are even functions

23

If y = f(x) is an odd function, then which of the following functions are also odd?

(i) Kf(x) (ii) f(x + K) (iii) f(x) + K

(A) (i) only

(B) (ii) only

(C) (iii) only

(D) (i) and (ii) only

(E) (ii) and (iii) only

(F) All are odd functions

24

True or False: (Explain your answers.)

(1) Every line has two distinct intercepts.

(2) Every line can be written in slope – intercept form.

(3) Two lines with the same slope but different x – intercepts must have the same y – intercept.

(4) Two lines with different slopes and the same x – intercept must have the same y – intercept.

(5) If you know the x – intercept and the y – intercept you can determine the equation of the line.

25

The y-intercept of is 2f x a x b c

(A) .

(B) .

(C) .

(D) .

(E) .

0,c

20,a b

20, ab c

20,ab c

0,b c

26

Consider the function, . Assume a > 0.Which of the following statements are true?

Increasing the value of a will

2( ) ( 2) 5f x a x

(i) increase the minimum value of f

(ii) change the vertex

(iii) change the y–intercept

(iv) not change the minimum value of f

(A) (i) and (ii) only (D) All are true

(B) (i), (ii) and (iii) only (E) All are false

(C) (iii) and (iv) only

27

A horticulturist has determined that the number of inches a young oak tree grows in one year is a function of the annual rainfall, r, given by

What is the maximum number of inches a young oak can grow in a year? The number of maximum inches is:

20.01 0.1 2g r r r

(A)Less than 1 (D) between 1 and 2

(B) Between 2 and 3 (E) between 3 and 4

(C) Between 4 and 5

28

True or False: (Explain your answers.)

(1) All even degree polynomials are even functions.

(2) All polynomials have at least on real zero.

(3) All polynomials of odd degree have all real numbers as their domain.

(4) A polynomial of degree 3 can have 4 real zeros.

(5) All polynomials have a y – intercept.

(6) A polynomial of degree 4 has 4 real zeros.

29

The graph of the polynomial function, is given below. The constant term of is

p x p x

(A)Greater then zero

(B) Equal to zero

(C) Less then zero

(D)Cannot be determined from graph

30

31

If , which of the following rational functions have only one vertical asymptote?

(1) (2) (3) 1 2 2x

r xx a

0a

2 2x

r xx ax

3 2 2x a

r xx a

(A) . (1) only

(B) . (2) only

(C) . (3) only

(D) . (2) and (3) only

(E) . All of them

(F) . None of them

32

Which of the following functions could have the graph shown below?

(A) .

(B) .

(C) .

(D) .

(E) .

2 2

( )( )x cf x

x a x b

2( )( )

x xfx a x b

( )( )( )

x x cf xx a x b

2( )( )

x cf xx a x b

( )( )( )

x cf xx a x b

33

If P(x) is a polynomial of degree , then the degree

of the quotient of is

(A) Less than 2

(B) Exactly 2

(C) Greater than or equal to 2

(D) Exactly 3

(E) None of these.

3n

P xx a

Recall

P x R xQ xD x D x

Q(x) is called the quotient, D(x) is the divisor and R(x) is the remainder.

34

Given and a > 0, which of the following

statements are true? Changing a will

1f x

x a

(i) changes x–intercept

(ii) changes y–intercept

(iii) changes horizontal asymptote.

A. (i) and (ii) only

B. (I) and (iii) only

C. (ii) only

D. (iii) only

E. None of these.

35

Match the function in Column I with the appropriate statement in Column II.

Column I

(1)

(2)

(3)

(4)

(5)

Column II

(A) The x-intercept is

(B) The vertical asymptote is

(C) The horizontal asymptote is y = 4

(D) The y intercept is

(E) There is a “hole’ in its graph at4( )1

xf xx

2 16( )4

xf xx

4 5( )2

xf xx

1( )4

f xx

8( )2

xf xx 4,0

4x

0, 4

4x

36

An insect population is increasing according to the exponential function defined by P = 3e0.03t, where P is in thousands and t is the number of years since 2004. Match the question given in Column I with the correct equation given in Column II.

Column I

(A)How long will it take the insect population to double?

(B) When will the population reach 5 thousand?

(C) How large will be the insect population be in the year 2007?

(D)How large is the population 3 months since 2004?

Column II

(1.)

(2.)

(3.)

(4.)

0.03 1/43P e

0.03 33P e

50.03

3t

e

60.03

3t

e

37

Compare the quantity in Column A with the quantity in Column B given the condition:

Column A Column B1x

2x 2x

(A)The quantity in Column A is greater

(B) The quantity on Column B is greater

(C) The two quantities are equal

(D)The relationship cannot be determined from the information given

38

Mia has an investment that doubles in value every 7 years. What will be the value of the investment 21 years after it has a value of $10,000?

(A) $50,000

(B) $60,000

(C) $70,000

(D) $80,000

(E) $90,000

39

Below is the graph of y = f(x) whereWhich of the following must be true?

xf x C b

(i)

(ii)

(iii)

(iv)

(A) (i) and (iii) only

(B) (i) and (iv) only

(C) (ii) and (iii) only

(D) (ii) and (iv) only

0C

0C

0 1b

1b

40

Which of the following functions have an inverse?

1.

2.

3.

f x x x

3 2

1

Tg T

T

2, for u t a t h k t h

(A)1 only

(B) 2 only

(C) 3 only

(D)1 and 3 only

(E) 2 and 3 only

(F) All have inverses

41

If f(x)=log(x+a) has a value of 1 when x = 1, then a must equal

(A) – 1

(B) e

(C) 9

(D) 10

(E) Not possible to determine with the given information.

(F) None of these.

42

If , what is the value of ? log 7xa 3log axa

(A) .

(B) 343 a

(C) a4

(D) 22

(E) None of these.

37a

43

What is the domain of ?

What is the domain of ?

What implications does your answers to the above have with regard to the solution of the equation

log 2 54

f x x

2log 3 log 2 54 4

x x

2log 34

x

44

Critique the following “solution” to the equation

2 2log log 6 4x x

2

2

2

4

2

2

log log 6 4

log 6 4

6 2

6 166 16 0

8 2 0

x x

x x

x x

x xx xx x

8 or 2x x

45

Do you expect the graph of

and the graph of

to be identical? Explain.

2logb

f x x

2logb

g x x

Do you expect the graph of

and the graph of

to be identical? Explain.

logb

f x x

logb

g x x

46

Let b be a positive real number (not equal to 1) and let a be its reciprocal.

Discuss the relationship between the graphs of

and logb

f x x logag x x

47

True / False. The functions represented by the graphs below are inverse functions.

2logy x

1/ 2logy x

48

Discuss why stretching / shrinking and vertical translations are the same for logarithmic functions.

That’s is, consider the transformation

where and

y f ax

logb

f x x , 0a x

49

y f x

1y f x

,a b

,b a 1 1f f x f f x x

50

Dow Jones Industrtial Avg. (Monthly Closing Prices)

0

2000

4000

6000

8000

10000

12000

14000

1600010/1/28

10/1/32

10/1/36

10/1/40

10/1/44

10/1/48

10/1/52

10/1/56

10/1/60

10/1/64

10/1/68

10/1/72

10/1/76

10/1/80

10/1/84

10/1/88

10/1/92

10/1/96

10/1/00

10/1/04

51

Dow Jones Industrial Averages (Monthly Closing Prices - Log Scale)

1

10

100

1000

10000

10000010/1/28

10/1/32

10/1/36

10/1/40

10/1/44

10/1/48

10/1/52

10/1/56

10/1/60

10/1/64

10/1/68

10/1/72

10/1/76

10/1/80

10/1/84

10/1/88

10/1/92

10/1/96

10/1/00

10/1/04