Deck 1: Functions, Graphs, and Limits

Question 1

Compute the indicated value of the given function. $$f ( x ) = 2 x ^ { 2 } - x$$ ; f (-1)

Answer 1

3

Question 2

Compute the indicated value of the given function. $$f ( x ) = 5 x ^ { 2 } - 3$$ ; f (-1)

Answer 2

2

Question 3

Compute the indicated values of the given function. f (x) = x - 3 + | x - 5| ; f (-4), f (-3), f (-2).

A) 8, 8, 8
B) -2, -2, -2
C) -6, -4, -2
D) 2, 2, 2

Answer 3

2, 2, 2

Question 4

Compute the indicated value of the given function. f (x) = | x - 3 | + x - 5; f (2)

A) -4
B) 2
C) -2
D) 1

Question 5

Compute the indicated value of the given function. f (x) = | x + 2 | + 3| x - 1 |; f (-3)

A) 13
B) -133
C) 9
D) 23

Question 6

Compute the indicated value of the given function. $$f ( t ) = \left\{ \begin{aligned}- 3 t + 5 & \text { if } t < - 1 \\t ^ { 2 } + 6 & \text { if } t \geq - 1\end{aligned} \right.$$ ; f (-3)

A) 14
B) -4
C) 15
D) -3

Question 7

Compute the indicated value of the given function. $$f ( x ) = \left\{ \begin{aligned}x ^ { 2 } - 1 & \text { if } x < 1 \\3 & \text { if } x = 1 \\2 - 5 x & \text { if } x > 1\end{aligned} \right.$$ ; f (2)

Question 8

Compute the indicated value of the given function. $$f ( x ) = \left\{ \begin{aligned}x ^ { 2 } - 5 & \text { if } x < 1 \\4 & \text { if } x = 1 \\7 - 8 x & \text { if } x > 1\end{aligned} \right.$$ ; f (2)

Question 9

Determine the domain of the given function. $$f ( x ) = \frac { x ^ { 2 } - 25 } { x + 2 }$$

A) all real numbers x
B) all real numbers x except x = -2
C) all real numbers x except x = 5 and x = 8
D) all real numbers x except x = 5, x = -5, and x = -2

Question 10

Determine the domain of the given function. $$f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 3 x + 1$$

A) all real numbers x
B) all real numbers x except x = 1
C) all real numbers x except x = -1
D) all real numbers x except x = 0

Question 11

Determine the domain of the given function. $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 } + 4 x + 1$$

Question 12

Determine the domain of the given function. $$f ( x ) = \sqrt { 2 x - 3 }$$

Question 13

Determine the domain of the given function. $$f ( x ) = \sqrt { x - 3 }$$

A) all real numbers x
B) all real numbers x except x = 3
C) all real numbers x for which $x \geq 3$
D) all real numbers x for which $x \geq - 3$

Question 14

Determine the domain of the given function. $$f ( t ) = \frac { \sqrt { t ^ { 2 } - 1 } } { t - 3 }$$

Question 15

Determine the domain of the given function. $$f ( t ) = \frac { \sqrt { t ^ { 2 } - 4 } } { t - 6 }$$

Question 16

Find the composite function $f ( g ( x ) )$ $f ( u ) = u ^ { 3 } + 2 , g ( x ) = x + 3$

A) $f ( g ( x ) ) = x ^ { 3 } + 5$
B) $f ( g ( x ) ) = x ^ { 3 } + 6$
C) $f ( g ( x ) ) = x ^ { 3 } + 9 x ^ { 2 } + 27 x + 27$
D) $f ( g ( x ) ) = x ^ { 3 } + x + 5$

Question 17

Find the composite function $f ( g ( x ) )$ . $f ( u ) = \frac { 1 } { u } - u$ , g(x) = 2x - 3

Question 18

Find the composite function $f ( g ( x ) )$ $f ( u ) = \frac { 1 } { u } , g ( x ) = x + 3$

A)
$f ( g ( x ) ) = \frac { 1 } { x } + 3$
B)
$f ( g ( x ) ) = x + 3 + \frac { 1 } { x }$
C)
$f ( g ( x ) ) = 1 + \frac { 3 } { x }$
D)
$f ( g ( x ) ) = \frac { 1 } { x + 3 }$

Question 19

Find the difference quotient, $\frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = \frac { 5 } { x }$

A)
$\frac { f ( x + h ) - f ( x ) } { h } = 1$
B)
$\frac { f ( x + h ) - f ( x ) } { h } = 5$
C)
$\frac { f ( x + h ) - f ( x ) } { h } = - \frac { 5 } { x ( x + h ) }$
D)
$\frac { f ( x + h ) - f ( x ) } { h } = \frac { 5 } { x + h }$

Question 20

Find the indicated composite function. f (x + 5) where $f ( x ) = \frac { 1 } { x }$

A)
$f ( x + 5 ) = \frac { 1 } { x + 5 }$
B)
$f ( x + 5 ) = \frac { 1 } { x } + 5$
C)
$f ( x + 5 ) = x + 5 + \frac { 1 } { x }$
D)
$f ( x + 5 ) = 1 + \frac { 3 } { x }$

Question 21

Find the indicated composite function. f (4x - 5) where $f ( x ) = \frac { 1 } { x } - x$

Question 22

Find functions h(x) and g(u) such that $f ( x ) = g ( h ( x ) )$ : $f ( x ) = \sqrt [ 7 ] { 84 - 7 x - 4 x ^ { 2 } }$

Question 23

Find functions h(x) and g(u) such that $f ( x ) = g ( h ( x ) )$ : $f ( x ) = \frac { 5 } { \sqrt { x + 4 } } + 3 \sqrt { x + 4 }$

Question 24

To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time required for the rat to traverse the maze on the nth trial was approximately $T ( n ) = 5 + \frac { 2 } { n } - \frac { 4 } { n ^ { 2 } }$ minutes. How many minutes does it take the rat to traverse the maze on the 2nd trial?

Question 25

To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time required for the rat to traverse the maze on the $n ^ { t h }$ trial was approximately $T ( n ) = 8 + \frac { 4 } { n } - \frac { 8 } { n ^ { 2 } }$ minutes. How long does it take the rat to traverse the maze in the second trial?

Question 26

A ball is thrown upward in such a way that t seconds later, it is $H ( t ) = - 16 t ^ { 2 } + 64 t + 80$ feet above the ground. How many seconds later does the ball hit the ground?

Question 27

A ball is thrown upward in such a way that t seconds later it is $H ( t ) = - 11 t ^ { 2 } + 44 t + 55$ feet above the ground. When does the ball hit the ground?

Question 28

At a certain factory, the total cost of manufacturing q units during the daily production run is $C ( q ) = q ^ { 2 } + 2 q + 297$ dollars. On a typical workday, q(t) = 17t units are manufactured during the first t hours of a production run. How many dollars are spent during the first 3 hours of production?

Question 29

At a certain factory, the total cost of manufacturing units during the daily production run is $C ( q ) = q ^ { 2 } + 2 q + 254$ dollars. On a typical day, q(t) = 16t units are manufactured during the first hours of a production run. How much is spent during the first 3 hours of production?

Question 30

Plot the given point in a rectangular coordinate system. (4, -5)

A) (Each gridline represents one unit.)
B) (Each gridline represents one unit.)
C) (Each gridline represents one unit.)
D) (Each gridline represents one unit.)

Question 31

Find the distance between the given points. (-2, 1) and (7, -5)

A) $D = \sqrt { 15 }$
B) D = 15
C) D = 117
D) $D = 3 \sqrt { 13 }$

Question 32

Sketch the graph of the given function. $f ( x ) = x ^ { 2 } + 2$ (Tick marks are spaced one unit apart.)

A)

B)

C)

D)

Question 33

Find the points of intersection (if any) of the given pair of curves. y = 3x - 1 and y = 3 - 5x

Question 34

Find the points of intersection (if any) of the given pair of curves. y = x + 2 and y = 2x + 4

A) (-2, 0)
B) (1, -4)
C) (0, 6)
D) (0, 2)

Question 35

Find the points of intersection (if any) of the given pair of curves. y = 7x - 4 and y = 3x - 5

Question 36

Find the points of intersection (if any) of the given pair of curves. $y = x ^ { 2 }$ and y = 3x - 2.

A) (0,0)
B) There are no points of intersection.
C) (2, 4)
D) (1, 1), (2, 4)

Question 37

If an object is thrown vertically upward with an initial speed of 128 ft/sec, its height (in feet) t seconds later is given by $H ( t ) = - 16 t ^ { 2 } + 128 t$ . Graph the function H(t) and use the graph to determine when the object will hit the ground.

Question 38

A ball is thrown upward in such a way that t seconds later, it is $s = - 16 t ^ { 2 } + 96 t + 144$ feet above the ground. Sketch the graph of s(t) and determine the maximum height in feet attained by the ball.

Question 39

Find the slope (if possible) of the line that passes through the given pair of points. (-1, 5) and (7, 2)

Question 40

Find the slope (if possible) of the line that passes through the given pair of points. (5, 0) and (3, 4)

A) -2
B) 2
C) $- \frac { 1 } { 2 }$
D) $\frac { 1 } { 2 }$

Question 41

Find the slope (if possible) of the line that passes through the given pair of points. (11, 0) and (14, 11)

A)
$\frac { 11 } { 3 }$
B)
$\frac { 3 } { 11 }$
C)
$- \frac { 3 } { 11 }$
D)
$- \frac { 11 } { 3 }$

Question 42

Find the slope (if possible) of the line that passes through the given pair of points. (19, 10) and (5, 8).

A) 7
B) $\frac { 1 } { 7 }$
C) 0
D) The slope is undefined.

Question 43

Find the slope (if possible) of the line that passes through the given pair of points. (6, 7) and (9, -1).

Question 44

Find the slope of the line passing through (4, 3) and (0, 4). Round your answer to two decimal places, if necessary.

A) -0.25
B) -4.00
C) 4.00
D) 0.25

Question 45

Find the slope and y-intercept of the line whose equation is given. 5y = 8x

A) slope 8, y-intercept 0
B) slope 8, y-intercept 5
C) slope $\frac { 8 } { 5 }$ , y-intercept 0
D) slope $- \frac { 8 } { 5 }$ , y-intercept 0

Question 46

Find the slope and y-intercept (if they exist) of the line 2y = 9x.

A) Slope is $\frac { 9 } { 2 }$ and y-intercept is 0.
B) Slope is 9 and y-intercept is 2.
C) Slope is 9 and y-intercept is 0.
D) Slope is $\frac { 2 } { 9 }$ and y-intercept is 0.

Question 47

Find the slope and y-intercept of the line whose equation is given. y = 2x - 5.

A) slope 2, y-intercept -5
B) slope 2, y-intercept 5
C) slope $\frac { 1 } { 2 }$ , y-intercept -5
D) slope $\frac { 1 } { 2 }$ , y-intercept 5

Question 48

Find the slope of the line whose equation is given. 3x + 5y = -7

Question 49

Find the slope of the line whose equation is given. 3x - 5y = 9.

Question 50

What is the slope of the line 7x + 8y = -3? Round your answer to two decimal places, if necessary.

A) -1.14
B) 1.14
C) 0.88
D) -0.88

Question 51

Find the slope of the line with the following equation 9x + 4y = -8.

Question 52

Find the slope and y-intercept of the line 3y - 7x = 1.

Question 53

Find the slope and y-intercept of the line whose equation is given. $\frac { x } { 3 } + \frac { y } { 5 } = 1$

Question 54

Find the slope and y-intercept of the line whose equation is given. $\frac { x } { 2 } + \frac { y } { 3 } = 1$

A) Slope: $- \frac { 3 } { 2 }$ ; y-intercept: 3
B) Slope: $\frac { 3 } { 2 }$ ; y-intercept: 1
C) Slope: $\frac { 1 } { 2 }$ ; y-intercept: 1
D) Slope: $\frac { 3 } { 2 }$ ; y-intercept: $\frac { 1 } { 2 }$

Question 55

Write an equation for the line through (3, 0) with slope 2.

A) y = 2x - 6
B) y = 2x - 3
C) y = 2x + 6
D) y = 2x + 3

Question 56

Write an equation for the line with the given properties. Through (3, -1) with slope 2

Question 57

Find an equation of the line parallel to the line y-axis and through the point (3, 9).

Question 58

What is the equation of the line through the points (7, -3) and (5, 19)? Round numbers to two decimal places, if necessary.

A) y = -0.09x + 74
B) y = -0.09x + 0.01
C) y = -11x + 0.01
D) y = -11x + 74

Question 59

Between August 29 and September 4, 2005, the price of gasoline increased steadily due to the impact of hurricane Katrina. The average price per gallon in Cincinnati on August 29 was $2.60, and on September 4 if was $3.10. Write a linear function C(x) that describes the cost per gallon of gas in Cincinnati in terms of days after August 29. Round the slope to three decimal places, if necessary. Use your function to find how much gas would have cost on October 1 if the price had continued to increase at that rate.

Question 60

The cost of renting a backhoe at one distributor is $335, plus $35 per day. Write a linear function C(x) that describes the cost of renting the backhoe for x days, then use your function to find how much it would cost to rent it for 7 days.

A) $C ( x ) = 335 x + 35 ; \quad \$ 2,380$
B) $C ( x ) = 7 ( 335 + 35 x ) ; \quad \$ 4,060$
C) $C ( x ) = 335 + 35 x , \quad \$ 580$
D) $C ( x ) = 35 x + 328 ; \quad \$ 573$

Question 61

Each unit of a certain commodity costs p = 11x + 12 cents when x units of the commodity are produced. If all units are sold at this price, express the revenue derived from the sales as a function of x.

A) x(11x + 12) cents
B) 10x + 12 cents
C) $11 x ^ { 2 } + 12$ cents
D) 12x + 12 cents

Question 62

A manufacturer's total cost consists of a fixed overhead of $250 plus production costs of $70 per unit. Express the total cost in dollars as a function of the number of units produced.

Question 63

A manufacturer's total cost consists of a fixed overhead of $350 plus production costs of $60 per unit. Express the total cost in dollars as a function of the number of units produced.

Question 64

A farmer is planning to plant a rectangular garden with an area of 4,000 square yards. The garden is to be fenced on all four sides. Express the number of yards of fencing required as a function of x, the long side of the fence.

Question 65

A farmer is planning to plant a rectangular garden with an area of 400 square yards. The garden is to be fenced on all four sides. Express the number of yards of fencing required as a function of x, the long side of the fence.

Question 66

A rectangle is constructed inside a circle of radius r with the corners of the rectangle lying on the circle. Express the area A of the rectangle as a function of the radius r if the width of one side of the rectangle is 2.

Question 67

A cylindrical can is to have a volume of 36 $\pi$ cubic inches. The cost of the material used for the top and bottom of the can is 4 cents per square inch, and the cost of the material used for the curved side is 3 cents per square inch. Express the cost in cents of constructing the can as a function of its radius.

Question 68

A cylindrical can is to have a volume of 45 $\pi$ cubic inches. The cost of the material used for the top and bottom of the can is 4 cents per square inch, and the cost of the material used for the curved side is 5 cents per square inch. Express the cost of constructing the can as a function of its radius.

Question 69

A closed box with a square base is to have a volume of 20 cubic meters. The material for the top and bottom of the box costs $3 per square meter, and the material for the sides costs $1 per square meter. Express the construction cost of the box as a function of the length of its base, x.

A) $C ( x ) = x ^ { 2 } + 80 x + 2$ dollars
B) $C ( x ) = 2 x ^ { 2 } + \frac { 2 } { x } + 6$ dollars
C) $C ( x ) = 2 x ^ { 2 } + \frac { 20 } { x }$ dollars
D) $C ( x ) = 6 x ^ { 2 } + \frac { 80 } { x }$ dollars

Question 70

The supply and demand functions, S(x) and D(x), respectively, for a certain commodity are given in terms of the level of production x. S(x) = 2x + 30, D(x) = 360 - x. Find the value of x_e for which equilibrium occurs.

Question 71

Two car rental agencies are competing. One agency rents cars for 35 dollars per day and 35 cents a mile; the other agency rents cars for 30 dollars per day and 50 cents a mile. For a 9 day trip, how many miles must you travel to have the total cost be the same with each agency? Round to the nearest whole mile , if necessary.

A) 300 miles
B) 33 miles
C) 14 miles
D) 10 miles

Question 72

Two jets bound for San Diego leave Boston 20 minutes apart. The first travels at 450 miles per hour, while the second goes 500 miles per hour. How how many hours after the second plane departs will the second plane pass the first?

Question 73

Two jets bound for San Diego leave Boston 20 minutes apart. The first plane travels at 300 miles per hour, while the second plane goes 320 miles per hour. At what time will the second plane pass the first?

Question 74

An appliance manufacturer can sell refrigerators for $600 apiece. The manufacturer's total cost consists of a fixed overhead of $12,000 plus production cost of $400 per refrigerator. How many refrigerators must be sold for the manufacturer to break even?

Question 75

An appliance manufacturer can sell refrigerators for $600 apiece. The manufacturer's total cost consists of a fixed overhead of $12,000 plus production cost of $400 per refrigerator. How many refrigerators must be sold for the manufacturer to break even?

Question 76

If f (x) is graphed below, find $\lim _ { x \rightarrow 1 } f ( x )$ if it exists.

Question 77

If f (x) is graphed below, find $\lim _ { x \rightarrow 1 } f ( x )$ if it exists.

Question 78

Find the indicated limit if it exists. $\lim _ { x \rightarrow 1 } \left( 2 x ^ { 2 } - 3 x + 1 \right)$

Question 79

Find the indicated limit if it exists. $\lim _ { x \rightarrow 1 } \left( 3 x ^ { 2 } - 8 x + 1 \right)$

Question 80

Find the indicated limit if it exists. $\lim _ { x \rightarrow 3 } ( x + 4 )$

A) 7
B) 3
C) Does not exist
D) -1