Deck 1: Functions and Models

Question 1

Find the value of $\log _ { 16 } 8$ .

A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 3 } { 4 }$
D)1
E) $\frac { 3 } { 2 }$
F)2
G)3
H)4

Answer 1

$\frac { 3 } { 4 }$

Question 2

(a) Sketch the curves represented by:
(i) x = t/2, y = (ii) x = sin t, y = 1 sin t
(iii) x = e , y = 1 - e (b) Describe the differences between the three curves in part (a).(c) Produce Cartesian equations for the curves in parts (a)(i), (ii), and (iii) by eliminating the parameter. Compare your results.

Answer 2

(b) (i) This curve represents the entire parabola y = 1 .
(ii) This curve represents the part of the parabola in (i) with x [ 1, 1].
(iii) This curve represents the part of the parabola in (i) with 0 x.
(c) They have the same Cartesian equation, y = 1 x with different domains.

Question 3

Find the inverse function for f (x) = $\frac { x - 1 } { x + 1 }$ .

A) $\frac { x + 1 } { x - 1 }$
B) $\frac { x } { x + 1 }$
C) $\frac { x + 1 } { x }$
D) $\frac { 1 + x } { 1 - x }$
E) $\frac { x + 1 } { 1 - x }$
F) $\frac { x } { x - 1 }$
G) $\frac { x - 1 } { x + 1 }$
H) $\frac { x - 1 } { x }$

Answer 3

$\frac { 1 + x } { 1 - x }$

Question 4

Find the domain of the inverse for f (x) = $\sqrt { 2 x - 5 }$ .

A)( $- \infty , - \frac { 5 } { 2 }$ ]
B)( $- \infty$ , 0]
C) $\left[ - \frac { 5 } { 2 } , \frac { 5 } { 2 } \right]$
D) $\left( - \infty , \frac { 5 } { 2 } \right]$
E) $\left[ - \frac { 5 } { 2 } , \infty \right)$
F)[0, $\infty$ )

G) $\left[ \frac { 2 } { 5 } , \infty \right)$
H) $\left[ \frac { 5 } { 2 } , \infty \right)$

Question 5

Eliminate the parameter in the equations $x = t ^ { 2 } , y = t ^ { 4 }$

A) $y = x ^ { 2 } \text { for } x \geq 0$
B) $y = \sqrt { x } \text { for } x \geq 0$
C) $y = 2 x ^ { 2 } \text { for } x \geq 0$
D) $y = \sqrt { 2 x } \text { for } x \geq 0$
E) $y = 2 \sqrt { x } \text { for } x \geq 0$
F) $y = x ^ { 2 } / 2 \text { for } x \geq 0$
G) $y = \sqrt { x } / 2 \text { for } x \geq 0$
H) $y = \sqrt { x / 2 } \text { for } x \geq 0$

Question 6

A baseball slugger hits a knee-high pitch toward the outfield. Suppose that the position of the baseball after t seconds is given by :
x = (v cos ) t, y = (v sin ) t where v is the velocity in feet per second at which the ball leaves the bat at an angle to the horizontal and from a height h above the ground.(a) Suppose that the ball is struck 2 feet above the ground with an initial velocity of 120 ft/sec and at an angle of 35 degrees.(i) When will the ball strike the ground?
(ii) How far will the ball travel (horizontally) before it touches the ground?
(iii) What is the maximum height reached by the ball?
(iv) Will the ball clear the 10 ft tall outfield fence, which is 410 feet away from the point where the ball is struck?
(b) If the ball is struck 2 feet above the ground at an initial velocity of 120 ft/sec and at an angle of 55 degrees:
(i) When will the ball strike the ground?
(ii) How far will the ball travel (horizontally) before it touches the ground?
(iii) What is the maximum height reached by the ball?
(iv) Will the ball clear the 10' tall outfield fence, which is 410 feet away from the point where the ball is struck?

Question 7

Consider the pairs of parametric equations and (a) Show that these pairs of equations produce the same line.(b) What are the slope and y-intercept of this line?

Question 8

Find the range of the inverse for f (x) = $- \frac { 3 } { 5 + 2 x }$ .

A) $\left( - \infty , - \frac { 5 } { 2 } \right)$
B)( $- \infty$ , 0)
C) $\left( - \frac { 5 } { 2 } , \frac { 5 } { 2 } \right)$
D) $\left( - \infty , \frac { 5 } { 2 } \right)$ $\cup$ $\left( \frac { 5 } { 2 } \infty \right)$
E) $\left( - \frac { 5 } { 2 } , \infty \right)$
F)(0, $\infty$ )

G) $\left( \frac { 5 } { 2 } , \infty \right)$
H) $\left( - \infty , - \frac { 5 } { 2 } \right)$ $\cup$ $\left( - \frac { 5 } { 2 } \infty \right)$

Question 9

Find the value of log $2$ $\frac { 1 } { 8 }$ .

A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 3 }$
C)0
D)1
E) - 1
F)2
G) - 2
H) - 3

Question 10

The position after t seconds of a projectile red with initial velocity v0 (measured in ft/s) at an angle above the horizontal from an initial height of h0 (measured in ft) is given by the parametric equations $x = \left( v _ { 0 } \cos \alpha \right) t$ $y = \left( v _ { 0 } \sin \alpha \right) t - 16 t ^ { 2 } + h _ { 0 }$ .
(a) Eliminate the parameter t to show that the trajectory of the projectile is a parabola.
(b) If the projectile is red with an initial velocity of 300 ft/s from a height of 5 ft, what should be the angle of elevation in order to hit a target at the same height as the initial height, 900 ft downrange?
(c) If the projectile is red with an angle of elevation of 40 ^$\circ$ and it strikes a target at the same height as the initial height, 600 ft downrange, what was the initial velocity of the projectile?

Question 11

Sketch a possible parametric curve defined by the following table of values.

Question 12

Given the function tan x with domain $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ find the domain of its inverse.

A) $\left[ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right]$
B)[0, $\infty$ )
C)[ $- \pi , \pi$ ]
D)[ $- 1,1$ ]
E) $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$
F) $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$
G)( $- \infty , \infty )$
H) $\left[ - \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right]$

Question 13

Suppose that the position of one particle at time t is given by
x = 4 sin t, y = 2 cos t for 0 and the position of a second particle is given by
x = 2 cos t 1, y = 4 sin t + 2 for 0 (a) How many points of intersection are there for these two paths?
(b) Are any of these collision points (that is, points where the particles are at the same place at the same time)? If so, find the collision points.

Question 14

At how many places does the curve x = cos 3t, y = sin t cross the x-axis?

A)1
E)5
B)2
F)6
C)3
G)7
D)4
H)8

Question 15

Eliminate the parameter in the equations $2 = \tan t , y = 2 \cos ^ { 2 } t , 0 \leq t \leq 2 \pi$

A) $y = 2 x ^ { 2 }$
B) $y = x ^ { 2 } + 4$
C) $y = 8 x ^ { 2 } + 32$
D) $y = \frac { x ^ { 2 } + 4 } { 8 }$
E) $y = x ^ { 2 } + 32$
F) $y = 4 x ^ { 2 } + 8$
G) $y = 8 x ^ { 2 } + 4$
H) $y = \frac { 8 } { x ^ { 2 } + 4 }$

Question 16

Describe the curve defined by x = cos t, y = sin $2$ t.

A)Circle
B)Semicircle
C)Quarter-circle
D)Parabola
E)Portion of a parabola
F)Hyperbola
G)Single branch of a hyperbola
H)Portion of branch of hyperbola

Question 17

Describe the motion of a particle with position x = 2 + cos t, y = 3 + sin t as t varies in the interval [0, 2 ].

Question 18

Find the inverse function for f (x) = $\frac { 5 - 2 x } { 3 }$ .

A) $\frac { 5 - 3 x } { 2 }$
B) $\frac { 2 x - 5 } { 3 }$
C) $\frac { 2 - 3 x } { 5 }$
D) $\frac { 3 x - 2 } { 5 }$
E) $\frac { 2 x + 5 } { 3 }$
F) $\frac { 3 } { 5 - 2 x }$
G) $\frac { 2 } { 5 - 3 x }$
H) $3 x - 5$

Question 19

Sketch a possible parametric curve defined by the following table of values.

Question 20

Find the value of log $1 / 2$ 1.

A) - 1
B) - $\frac { 1 } { 2 }$
C)0
D)10 $2$
E)1
F) $\frac { 1 } { 2 }$
G)2

H) - 2

Question 21

Find the value of (4 ln 5 ln 1)(ln ).

Question 22

Solve the equation $\log _ { 9 }$ (ln $x ^ { 3 }$ ) = 1

A) $x = 3 ^ { e }$
B) $x = 3 e$
C) $x = e / 3$
D) $x = 1$
E) $x = e ^ { 2 }$
F) $x = 1 / e$
G) $x = 3 / e$
H) $x = e ^ { 3 }$

Question 23

Find the value of $e ^ { 3 \ln 2 }$ .

A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 2 }$
C)5
D)6
E)8
F)9
G)12
H)18

Question 24

Sketch the graph of f for f (x) = and determine if f exists . If so, find a formula for f (x) and sketch its graph.

Question 25

Find the value of ln e $e$ .

A) $- 1$
B) $1 / \sqrt { e }$
C)e
D)0
E) $\sqrt { e }$
F)1/e
G)1
H) - e

Question 26

Find the value of 4

Question 27

Solve for ln .

Question 28

Find the value of ln $\sqrt { e ^ { 3 } }$ .

A) $\frac { 2 } { 3 }$
B) $\sqrt { e }$
C) $e ^ { 3 } / 2$
D) $\frac { 3 } { 2 }$
E) $e ^ { 3 }$
F) $e ^ { 3 }$ $- 2$
G) $2 e / 3$
H)2/e $3$

Question 29

Find if .

Question 30

Express 2 log x 5 log (x 10) as a single logarithm.

Question 31

Suppose pH = -log [H ]. Suppose further that for vinegar, the hydrogen ion concentration in moles per liter is given by [H ] . Find the pH of the vinegar.

Question 32

Solve the equation $e ^ { 2 - 3 x } = 125$ .

A) $x = 2 - \ln 5$
B) $x = 2 - 3 \ln 5$
C) $x = \frac { 2 } { 3 } - \ln 5$
D) $x = \frac { 2 } { 3 } - 3 \ln 5$
E) $x = - \ln 5$
F) $x = - \frac { 1 } { 3 } \ln 5$
G) $x = \frac { 2 } { 3 } \ln 5$
H) $x = 2 + 3 \ln 5$

Question 33

Determine whether or not the function f (x) = is one-to-one.

Question 34

Find the value of $\log _ { 2 } e - \log _ { 2 } ( e / 16 )$ .

A) $- 2$
B) $e ^ { - 2 }$
C)4
D) $e ^ { 16 }$
E) $- 4$
F) $e ^ { 2 }$
G)2

H) $e ^ { - 16 }$

Question 35

Determine whether or not the function f (x) = x 2x + 5 is one-to-one.

Question 36

Solve for .

Question 37

Solve the equation $e ^ { 2 x - 2 } = 4$ .

A) $x = \ln 2$
B) $x = 1 - \ln 2$
C) $x = 1 + \ln 2$
D) $x = 1 - 2 \ln 2$
E) $x = 1 + 2 \ln 2$
F) $x = 2 + \ln 2$
G) $x = 2 - \ln 2$
H) $x = 2 - 2 \ln 2$

Question 38

Find the inverse of f (x) .

Question 39

Make a rough sketch of the graph of y = 2 + log (x + 1). Do not use a calculator. Instead, start with the graph of a simple function and apply any needed transformations.

Question 40

Make a rough sketch of the graph of y = 2 + log (x + 1). Do not use a calculator. Instead, start with the graph of a simple function and apply any needed transformations.

Question 41

Given the graph of y = 2 , find an equation of the graph that results from reflecting the given graph about
(a) the line x = 0.(b) the line y = 3.(c) the line y = -1.(d) the line x = 2.

Question 42

A bacteria culture starts with 500 bacteria and doubles every 2 hours. How many bacteria are there after 6 hours?

A)1,000
E)3,000
B)1,500
F)4,000
C)2,000
G)5,000
D)2,500
H)10,000

Question 43

For what value of x is 3 $4 - x$ = $\sqrt { 3 }$ ?

A)0
B) $\frac { 1 } { 2 }$
C)1
D) $\frac { 3 } { 2 }$
E)2
F) $\frac { 5 } { 2 }$
G)3

H) $\frac { 7 } { 2 }$

Question 44

Decide whether the given function has an inverse. Defend your decision.(a) P (x) is the cost, in cents, of mailing a letter that weighs x ounces.(b) C (t) is the total number of cars which have driven past a particular point along a highway during a specific day where t represents the time, in hours, since midnight.(c) F (d) is the amount of fuel, in gallons, in your car when you have traveled d miles on a particular tankful of gas.(d) S (t) is the number of shoppers entering the Mall of America where t is the number of minutes past midnight on one particular day.(e) T (t) is the temperature inside an oven where t is the time, in minutes, from the beginning to the end of the oven's self-cleaning cycle.(f) V (x) is the volume of a cube whose side has length x.(g) f (n) is the number of students enrolled in your calculus class on the nth day of the term.

Question 45

Suppose that the number of bacteria in a culture at time t is given by Use natural logarithms to solve for t in terms of x.

Question 46

Medical professionals sometimes use iodine-131, a radioactive substance, to diagnose certain conditions of the thyroid gland. The formula for the proportion P of iodine-131 remaining in a patient's system t days after receiving the substance is given by P .(a) Find the inverse of this function and explain its meaning.(b) How long does it take for the proportion to drop to 10% of the original dosage?

Question 47

Make a rough sketch of the graph of y = 3 (4 2 ). Do not use a calculator.

Question 48

Suppose that P(t) represents the pressure (in pounds per square inch) in a tire with a slow air leak t minutes after the leak begins. In practical terms, what is:
(a) P(20)
(b) P (20)

Question 49

If a function f has an inverse and is an increasing function, can you determine if the inverse is increasing or decreasing? Explain.

Question 50

Make a rough sketch of the graph of y = 4 + 1. Do not use a calculator.

Question 51

Which of the following statements are true about the graph of the function y = 2 $x$ - 8?
(i) It has no vertical asymptote.(ii) It has no horizontal asymptote.(iii) It has a y-intercept at -3.(iv) It has an x-intercept at 3.

A)(i) only
B)(i), (iii), and (iv) only
C)(ii), (iii), and (iv) only
D)(iv) only
E)(ii) and (iii) only
F)(i) and (iv) only
G)(ii) and (iv) only
H)(i), (ii) and (iv) only

Question 52

What single transformation of the graph of y = e is the same as shifting the graph of y = e four units downward and then reflecting the shifted graph in the x-axis?

Question 53

For the exponential function f (x) = c . a $x$ + b whose graph is given below, determine the values of a, b, and c.

Question 54

The following table defines the function f (x). From this table, write a table for f : Determine the domain for f .

Question 55

A bacteria culture starts with 200 bacteria and triples in size every ten minutes. After 1 hour, how many bacteria are there?

A)1800
E)48,600
B)3600
F)145,800
C)5400
G)437,400
D)16,200
H)1,312,200

Question 56

The half-life of a certain radioactive substance is 5 days. The initial size of a sample is 10 grams.(a) Find the amount of the substance remaining after 20 days.(b) Find the amount of the substance remaining after t days.(c) Use a graph to estimate, to the nearest 0:01 gram, the amount remaining after 14 days.(d) Use the graph to estimate, to the nearest 0:1 day, the amount of time required for the mass of the substance to be reduced to 0:1 gram.

Question 57

Make a rough sketch of the graph of y = -2 . Do not use a calculator.

Question 58

Make a rough sketch of the graph of y = (1.1) . Do not use a calculator.

Question 59

Make a rough sketch of the graph of y = 3 . Do not use a calculator.

Question 60

The radioactive isotope Bismuth-210 has a half-life of 5 days. How many days does it take for 87.5% of a given amount to decay?

A)15
E)11
B)8
F)9
C)10
G)12
D)13
H)14

Question 61

Determine an appropriate viewing window for each of the following functions and use it to draw the graph.(a) f (x) = x 6x (b) f (x) = (c) f (x) = sin (x )
(d) f (x) = 80x 5x (e) f (x) = 3x + 288x 6862
(f) f (x) = 3x + 288x + 6862

Question 62

Relative to the graph of y = sin x, the graph of y = 3 sin x is changed in what way?

A)Compressed horizontally by a factor of 3
B)Shifted 3 units to the right
C)Compressed vertically by a factor of 3
D)Shifted 3 units upward
E)Shifted 3 units to the left
F)Stretched vertically by a factor of 3
G)Shifted 3 units downward
H)Stretched horizontally by a factor of 3

Question 63

Determine an appropriate viewing rectangle for the graph of the function f (x) = 4x - $\left| 3 x ^ { 2 } - 10 \right|$ .

A)[ - 5, 5] $\times$ [ - 5, 2]
B)[ - 50, 20] $\times$ [ - 50, 10]
C)[ - 2, 5] $\times$ [ - 1, 10]
D)[ - 10, 10] $\times$ [ - 500, 500]
E)[ - 10, 10] $\times$ [ - 500, 5]
F)[ - 2, 5] $\times$ [ - 15, 10]
G)[ - 10, 5] $\times$ [ - 1, 10]
H)[0, 10] $\times$ [ - 2, 5]

Question 64

Determine the number of solutions of the equation 5x $4$ - 3x $2$ = 10x $5$ - 20x $6$ $+$ 63x $+$ 5.

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question 65

Relative to the graph of y = x $2$ $+$ 2, the graph of y = 4x $2$ $+$ 2 is changed in what way?

A)Compressed vertically by a factor of 2
B)Stretched horizontally by a factor of 2
C)Compressed horizontally by a factor of 2
D)Shifted 2 units upward
E)Shifted 2 units to the right
F)Stretched vertically by a factor of 2
G)Shifted 2 units to the left
H)Shifted 2 units downward

Question 66

Relative to the graph of y = x $3$ , the graph of y = $\frac { 1 } { 2 }$ x $3$ is changed in what way?

A)Compressed horizontally by a factor of 2
B)Shifted 2 units downward
C)Stretched vertically by a factor of 2
D)Stretched horizontally by a factor of 2
E)Shifted 2 units upward
F)Compressed vertically by a factor of 2
G)Shifted 2 units to the right
H)Shifted 2 units to the left

Question 67

Determine an appropriate viewing rectangle for the graph of the function $f ( x ) = ( x + 50 ) ^ { 2 } + 1000$ .

A)[ - 60, - 40] $\times$ [40, 60]
B)[2400,2600] $\times$ [900, 120]
C)[40,60] $\times$ [980,1020]
D)[40,60] $\times$ [ - 1020, - 980]
E)[40, 60] $\times$ [40, 60]
F)[ - 60, - 40] $\times$ [980, 1020]
G)[ - 10, 10] $\times$ [ - 10, 10]
H)[980, 1020] $\times$ [980, 1020]

Question 68

Let f(x) = 2 - x $3$ and g(x) = 3 $+$ x. Find the value of $( f \circ g ) ( x )$ when x = - 5.

A) - 510
B) - 5
C) - 2
D)0
E)5
F)10
G)127
H)130

Question 69

Match each set of function values in the table with the formula which best fits it. Assume that a, b, and c are constants.

Question 70

Determine an appropriate viewing rectangle for the graph of the function f (x) = 6x $2$ + x - 10.

A)[ - 50, 50] $\times$ [ - 50, 50]
B)[0, 10] $\times$ [0, 10]
C)[ - 5, 10] $\times$ [ - 10, 50]
D)[ - 5, 5] $\times$ [ - 5, 5]
E)[ - 5, 5] $\times$ [ - 10, 10]
F)[ - 50, 10] $\times$ [ - 10, 100]
G)[ - 5, 50] $\times$ [ - 5, 10]
H)[ - 10, 10] $\times$ [ - 5, 5]

Question 71

Determine an appropriate viewing rectangle for the graph of the function f (x) = 20 sin (40x) $+$ 10.

A)[ - 20, 20] $\times$ [ - 20, 20]
B)[ - 0.2, 0.2] $\times$ [ - 20, 20].
C)[ - 20, 20] $\times$ [ - 0.2, 0.2]
D)[ - 0.2, 0.2] $\times$ [ - 10, 30]
E)[ - 20, 20] $\times$ [ - 20, 20] .
F)[ - 0.2, 0.2] $\times$ [ - 1, 1]
G)[ - 10, 10] $\times$ [ - 10, 10]
H)[ - 1, 1] $\times$ [ - 10, 10].

Question 72

Relative to the graph of y = x $2$ + 2, the graph of y = (x - 2) $2$ $+$ 2 is changed in what way?

A)Shifted 2 units upward
B)Compressed vertically by a factor of 2
C)Compressed horizontally by a factor of 2
D)Shifted 2 units to the left
E)Shifted 2 units to the right
F)Shifted 2 units downward
G)Stretched vertically by a factor of 2
H)Stretched horizontally by a factor of 2

Question 73

Find the difference between the largest and smallest solutions of the equation - 3x $2$ - 9x = 2 $x$ , rounded to two decimal places.

A)0.02
B)0.23
C)0.80
D)1.05
E)2.06
F)2.88
G)3.09
H)3.92

Question 74

Let f(x) = 3x - 2 and g(x) = 2 - 3x. Find the value of (f ^$\circ$ g)(x) when x = 3..

A) - 23
B) - 9
C) - 6
D) - 3
E)3
F)6
G)9
H)23

Question 75

Relative to the graph of y = x $2$ , the graph of y = x $2$ - 2 is changed in what way?

A)Shifted 2 units downward
B)Stretched horizontally by a factor of 2
C)Shifted 2 units to the right
D)Stretched vertically by a factor of 2
E)Compressed horizontally by a factor of 2
F)Compressed vertically by a factor of 2

Question 76

Let h(x) = sin $2$ x $+$ 3 sin x - 4 and g(x) = sin x. Find f(x) so that h(x) = $( f \circ g ) ( x )$

A)f (x) = (3x + 2) $2$
B)f (x) = x + 3
C)f(x) = 3x $2$ - 4
D)f(x) = x $2$ - 3x $+$ 4
E)f (x) = 3x $2$ - 4x
F)f (x) = x $2$ $+$ 3x - 4
G)f (x) = x $2$ - 4
H)f (x) = (x - 4) $2$

Question 77

Consider the family of curves given by y = x + cx. Graph the function for values of c = 4, 2, 0, 2, and 4. What characteristics are shared by all of the graphs? How does changing the value of c affect the graph?

Question 78

Determine the number of real solutions of the equation 5x $4$ - 3x $2$ = 10x $5$ - 20x $3$ + 5.

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question 79

Let f(x) = $\frac { 1 } { 2 }$ x and $( f \circ g ) ( x ) = x ^ { 2 }$ . Find g(2).

A)0
B)1
C)2
D)4
E)8
F)16
G)32
H)64

Question 80

Determine an appropriate viewing rectangle for the graph of the function f (x) = $\frac { 3 x } { x ^ { 4 } + 75 }$ ..

A)[ - 10, 10] $\times$ [ - 10, 10]
B)[ - 1, 1] $\times$ [ - 1, 1]
C)[ - 10, 10] $\times$ [ - 1, 1]
D)[ - 100, 100] $\times$ [ - 1, 1]
E)[ - 100, 100] $\times$ [ - 10, 10]
F)[ - 5, 5] $\times$ [ - 5, 5]
G)[ - 5, 5] $\times$ [ - 0.5, 0.5]
H)[ - 1, 1] $\times$ [ - 10, 10]