LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME … A… · 5. Sketch the graphs of y = x2 − 9, y = 9− x2 , and y = 9− x2 on a single set of axes. 6. Sketch the graphs of

Trig/Math Anal Name_______________________No_____

- 1 -

Practice Set A (Pg. 122-124) Tell whether the graph of each relation is the graph of a function. If it is, give the domain and range of the function.

7. Explain why the equation 122 =+ yx does not define

y as a function of x .

8. Explain why the equation 133 =+ yx defines y

as a function of x .

Give the domain of each function

9a. ( )x

xf1

= 9b. ( )9

1

−=

xxg 9c. ( )

4

32 −

=x

xxh

10a. ( )3

1

+=

ttf 10b. ( )

65

22 ++

+=

tt

ttg 10c. ( )

tt

tth

9

323

2

−

−=

Give the domain, range, and zeros of each function.

LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON _____ NO GRAPHING CALCULATORS ALLOWED ON THIS TEST

HW NO. SECTIONS

(Brown book)

ASSIGNMENT DUE √

F-1 4-1 Practice Set A 1-9, 11-14, 17, 21, 22

F-2 4-2 Practice Set B 1-11, 13, 17, 19, 23, 25, 27-29, 31, 33, 35

F-3A 4-3

Reflections

Practice Set C 1-14

F-3B 4-3

Symmetry

Practice Set C 15, 27-34

F-4 4-4

(Pearson Book)

Practice Set D #7-18 all; 19-35 odd; 39-65 odd

F-5 4-4 Practice Set E #1-5, 7-11, 12a, 13a-d, 15

F-6 4-5 Practice Set F #1-3, 5-23

Practice Set D #20-34 even

F-7 4-7 Practice Set G 1-27 odd

F-8 Review Practice Set H #1-9


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11a. ( ) xxf = 11b. ( ) 2−= xxg 11c. ( ) 2−= xxh

12a. ( ) ttf = 12b. ( ) ttg −= 9 12c. ( ) 29 tth −=

Ex. 13-18, sketch the graph of each function. Use the graph to find the range and zeros of the function.

13. ( ) 862 +−= xxxf 14. ( ) ( )234 −−= xxg 15. ( ) ( )3

2−= ttf

16. ( ) 44 23 −−+= ttttg 17.

( )

<≤−

<≤−=

41 if 2

12 if 2

uu

uuuh 18. ( )

>

≤≤−−

<−

=

3 if 0

30 if 32

0 if 12

u

uuu

uu

ug

19 a. Let V be the function that assigns to each solid its volume. If C is a cylinder with radius 3 and height 4, find V(C).

b. Give the domain and range.

20 a. A formula from geometry states that ( 2)180.S n= − Give the meaning of this formula.

b. Is S a function of n ? If so, give the domain and range of this function.

21. The greatest integer function assigns each number the greatest integer less than or equal to the number. If

we denote the greatest integer in x by x , then we have 528.5 = , 55 = , 3=π , and 27.1 −=− .

a. Sketch the graph of xy =

b. Give the domain and range of the greatest integer function.

22. The greatest integer function ( ) xxf = is sometimes called the “floor of x”. By contrast, ( ) xxc = is

called the “ceiling of x” and is the least integer greater than or equal to x. Thus 628.5 = , 55 = , 4=π ,

and 17.1 −=− .

a. Sketch the graph of xy =

b. The cost of parking a car in a municipal parking lot is $3 for the first hour or any part thereof, plus $2 for each additional hour or part thereof. Sketch the graph of this cost function and find a rule for the cost C as a function of time t. Your rule should use the ceiling function.

23. Think about what it means for two functions to be equal. Would you say that the function ( )f x x= and

2( )g x x= are equal? Write a brief defense of your conclusion.

24. Use a mathematics dictionary to find the meaning of the phrase implicit function. Then determine what implicit functions, if any, are defined by each of the following equations.

a. 2x y= b. 2 2 1x y+ = c. 2 2 0x y− =

25. For which of the following functions does ( ) ( ) ( )f a b f a f b+ = + ?

a. 2( )f x x= b. 1

( )f xx

= c. ( ) 4 1f x x= + d. ( ) 4f x x=

26. For which of the functions in Exercise 25 does ( ) ( ) ( )f ab f a f b= ⋅ ?

Practice Set B (Pg. 128)

1. Copy the graph of ( )xfy = shown. On a single set of

axes, draw the graphs of ( ) 2+= xfy and ( ) 3−= xfy

2. On a single set of axes, graph xy = ,

5+= xy , and 4−= xy


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3. The graphs of ( )xf and ( )xg are shown below.

a. Find ( ) ( )11 gf −

b. For what values of x is ( ) ( )xgxf − positive? Negative?

Zero?

c. What is the maximum value of ( ) ( )xgxf − ?

4. The graphs of ( )xf and ( )xg are shown

below.

a. Find ( ) ( )11 gf −

b. For what values of x is ( ) ( )xgxf − positive?

Negative? Zero?

c. What is the maximum value of ( ) ( )xgxf − ?

Let ( ) 13 −= xxf and ( ) 1−= xxg . Evaluate the following expressions.

5. ( )( )xgf + 6. ( )( )xgf − 7. ( )( )xgf ⋅

8. ( )xg

f

9a. ( )( )2gf

9b. ( )( )xgf �

10a. ( )( )2fg

10b. ( )( )xfg �

11. On a single set of axes, graph ( ) xxf = in one color, graph ( ) xxg = in a second color, and graph gf + in

a third color.

12. Using the graphs given in Exercise 4, graph gf +

13. Given two functions, f and g , one way to obtain the real solutions of the equation ( ) ( )f x g x= is to graph

the equation ( )y f x= and ( )y g x= in an xy − plane and then find the x − coordinates of any points of

intersection. Describe another way to solve ( ) ( )f x g x= that also involves graphing in an xy − plane but that is

based on the difference function f g− .

Use a computer or graphing calculator and one of the methods from Exercise 13 to find the real solutions of each of the following equations. Give answers to the nearest hundredth.

14. 3 1x x= + 15. 1 2x x+ = 16. 21 x x− =

17. Let ( ) ( ) ( ) 23 and ,2

3 ,32 +=

+=−= xxh

xxgxxf

a. Show that ( )( ) ( )( )xfgxgf = for all x.

b. Show that ( )( ) ( )( )xfhxhf ≠ for any x.

18. Let ( ) ( ) ( ) xxhxxgxxf 3 and , , 33 ===

a. Show that ( )( ) ( )( )xfgxgf = for all x.

b. Show that ( )( ) ( )( )xfhxhf = for only one value of x.

Let ( ) ( ) ( )3

and ,36 ,x

xhxxgxxf =−== . Find each of the following.

19a. ( )( )( )6hgf 19b. ( )( )( )xhgf 20a. ( )( )( )4fgh 20b. ( )( )( )xfgh

21a.

2

1gfh

21b. ( )( )( )xgfh 22a. ( )( )( )9fhg 22b. ( )( )( )xfhg

Let ( ) ( ) ( ) ( ) xxjxxhxxgxxf 2 and ,4 , ,3 =−=== . Express each function k as a composite of three of these

four functions.


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23. ( ) ( )342 −= xxk 24. ( ) ( )3

4−= xxk 25. ( ) ( )382 −= xxk 26. ( ) 43 −= xxk

27. Physiology. The graph at the left below shows a swimmer’s speed s as a function of time t. The graph at the right below shows the swimmer’s oxygen consumption c as a function of s. Time is measured in seconds, speed in meters per second, and oxygen consumption in liters per minute. a. What are the speed and oxygen consumption after 20 s of swimming? b. How many seconds have elapsed if the swimmer’s oxygen consumption is 15 L/min?

28. Consumer Economics. The graph at the left below shows a car’s fuel economy e as a function of the speed s at which the car is driven. The graph at the right below shows the per-mile fuel cost c as a function of e. Fuel economy is measured in miles per gallon, speed in miles per hour, and fuel cost in cents per mile. a. If the car is driven at 55 mi/h, what is the fuel cost? b. If the fuel cost is to be kept at or below 4 cents per mile, at what speeds should the car be driven?

29. a. Express the radius r of a circle as a function of the circumference. b. Express the area A of the circle as a function of C.

30 a. Express the area A and perimeter P of a semicircular region as a function of the radius, r . b. Express A as a function of P.

31. Physics The speed s of sound in air is given by the

formula Cs 6.0331+= where s is measured in meters per second and C is the Celsius temperature. If

( )329

5−= FC , express s as a function of F, the

Fahrenheit temperature.

32. The surface area and volume of a sphere are given in terms of the radius by the following formulas:

24A rπ= and 34

3V rπ= .

a. Express r as a function of A

b. Express V as a function of A

In Exercises 33-36, find rules for ( )( )xgf � and ( )( )xfg � and give the domain of each composite function.

33. ( ) ( ) 216 ,2 xxgxxf −== 34. ( ) ( )4

1 ,

−==

xxgxxf

35. ( ) ( ) xxgxxf −== 1 ,2 36. ( ) ( ) 22 16 , xxgxxf −==

37. If ( )2

3+=

xxg , find

( )( ) ( )( )( ) ( )( )( )( )1 and ,1 ,1 ggggggggg .

38. If ( ) 12 −= xxf , show that ( )( ) 34 −= xxff . Find

( )( )( )xfff .

Practice Set C (Pg. 136)

In Exercises 1-4, the graph of ( )xfy = is given. Sketch the graphs of:

a. ( )xfy −= b. ( )xfy = c. ( )xfy −=


- 5 -

5. Sketch the graphs of 222 9 and ,9,9 xyxyxy −=−=−= on a single set of axes.

6. Sketch the graphs of xyxyxy −=−=−= 2 and ,2 ,2 on a single set of axes.

In Exercises 7-14, sketch the graph of each equation and the reflection of the graph in the line xy = . Then give

an equation of the reflected graph.

7. 43 −= xy 8. 1

2

1+= xy

9. xxy 22 −= 10. xxy 32 +=

11. 3xy = 12. xy = 13. 2+= xy 14. 3−= xy

15. Test each equation to see if its graph has symmetry in: (i) the x-axis, (ii) the y-axis, (iii) the line xy = and (iv) the origin

a. 22 =− xyy b. 122 =+ yx c. xxy =

16. Repeat Exercise 15 for each of the following equations.

a. 42 =+ xyx b. 1=+ yx c. x

xy =

In Exercises 21-26, graph each parabola, showing the vertex with its coordinates and the axis of symmetry with its equation. The pairs of graphs in Exercises 21-23 should be done on a single set of axes.

21. a. ( ) 532

+−= xy

b. ( ) 532

+−= yx

22. a. ( ) 3122

++= xy

b. ( ) 3122

++= yx

23. a. ( )243 −−= xy

b. ( )243 −−= yx

24. ( ) 4122

++= yx 25. 862 ++= yyx 26. 322 −+= yyx

27. The graph of a cubic function has a local minimum at (5, -3) and a point of symmetry at (0, 4). At what point does a local maximum occur?

28. a. Find the point of symmetry of the graph of the cubic function ( ) 454815 23 +−+−= xxxxf .

b. the function has a local minimum at (2, 1). At what point does a local maximum occur?

29. a. Graph 323 xxy −= . At what point does a local minimum occur?

b. Find the point of symmetry of the graph and then deduce the coordinates of the point where a local maximum occurs?

30. a. Graph xxxy 96 23 −−−= . At what point does a local minimum occur?

b. Find the point of symmetry and then deduce the coordinates of the point where a local maximum occurs.

Use the following definitions to complete Exercises 31-36.

f is an even function if ( ) ( )xfxf =−

f is an odd function if ( ) ( )xfxf −=−


- 6 -

31. Classify each function as even, odd, or neither.

a. ( ) 2xxf = b. ( ) 3xxf = c. ( ) xxxf −= 2

d. ( ) 24 2xxxf += e. ( ) 23 3xxxf += f. ( ) 35 4xxxf −=

32. Use the results of Exercise 31 to guess the reasons for using the terms “even” and “odd” as they are applied to polynomial functions.

33. a. What kind of symmetry does the graph of an even function have? b. What kind of symmetry does the graph of an odd function have?

34. Study the graphs shown in Exercises 3 and 4. Then tell whether each function graphed is even or odd.

Practice Set D (Pg. 112)

In problems 19-26, write the function whose graph is the graph of 3xy = , but is:

19. Shifted to the right 4 units 20. Shifted to the left 4 units

21. Shifted up 4 units 22. Shifted down 4 units

23. Reflected about the y-axis 24. Reflected about the x-axis

25. Vertically stretched by a factor of 4 26. Horizontally stretched by a factor of 4

In problems 27-30, find the function that is finally graphed after each of the following transformations is applied

to the graph of xy = in the order stated.

27. (1) Shift up 2 units (2) Reflect about the x-axis (3) Reflect about the y-axis

28. (1) Reflect about the x-axis (2) Shift right 3 units (3) Shift down 2 units


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29. (1) Reflect about the x-axis (2) Shift up 2 units (3) Shift left 3 units

30. (1) Shift up 2 units (2) Reflect about the y-axis (3) Shift left 3 units

31. If (3, 6) is a point on the graph of ( )xfy = ,

which of the following points must be on the graph of

( )xfy −= ?

a) (6, 3) b) (6, -3) c) (3, -6) d) (-3, 6)

32. If (3, 6) is a point on the graph of ( )xfy = , which

of the following points must be on the graph of

( )xfy −= ?

a) (6, 3) b) (6, -3) c) (3, -6) d) (-3, 6)



( )xfy 2= ?

a)

2

3,1 b) (2,3)

c) (1, 6) d)

3,

2

1



( )xfy 2= ?

a) (4,1) b) (8, 2) c) (2, 2) d) (4, 4)

35. Suppose that the x-intercepts of the graph of

( )xfy = are -5 and 3.

a) What are the x-intercepts of the graph of

( )2+= xfy ?

b) What are the x-intercepts of the graph of

( )2−= xfy ?

c) What are the x-intercepts of the graph of

( )xfy 4= ?

d) What are the x-intercepts of ( )xf − ?

36. Suppose that the x-intercepts of the graph of

( )xfy = are -8 and 1.

a) What are the x-intercepts of the graph of

( )4+= xfy ?

b) What are the x-intercepts of the graph of

( )3−= xfy ?

c) What are the x-intercepts of the graph of

( )xfy 2= ?

d) What are the x-intercepts of ( )xf − ?

In problems 39-66, graph each function using the techniques of shifting, compressing, stretching, and/or

reflecting. Start with the graph of the basic function (for example, 2xy = ) and show all stages. Find the

domain and range of the final function. Verify your results using a graphing utility.

39. ( ) 12 −= xxf 40. ( ) 42 += xxf 41. ( ) 13 += xxg

42. ( ) 13 −= xxg 43. ( ) 2−= xxh 44. ( ) 1+= xxh

45. ( ) ( ) 213

+−= xxf 46. ( ) 323

−+= xfx 47. ( ) xxg 4=

48. ( ) xxg = 49. ( )x

xh2

1= 50. ( ) 3 2xxh =

51. ( ) 3 xxf −= 52. ( ) xxf −= 53. ( ) 3 xxg −=

54. ( )x

xg−

=1

55. ( ) 23 +−= xxf

56. ( ) 21

+−

=x

xh

57. ( ) ( ) 3122

−+= xxf 58. ( ) ( ) 1232

+−= xxf 59. ( ) 122 +−= xxg

60. ( ) 313 −+= xxg 61. ( ) 2−−= xxh 62. ( ) 24

+=x

xh

63. ( ) ( ) 113

−+−= xxf 64. ( ) 14 −−= xxf 65. ( ) xxg −= 12

66. ( ) xxg −= 24


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In problems 69-72, the graph of a function f is illustrated. Use the graph of f as the first step toward graphing each of the following functions:

(a) ( ) ( ) 3+= xfxF (b) ( ) ( )2+= xfxG (c) ( ) ( )xfxP −= (d) ( ) ( ) 21 −+= xfxH

(e) ( ) ( )xfxQ2

1= (f) ( ) ( )xfxg −= (g) ( ) ( )xfxh 2=

Practice Set E (pg 143)

In Exercises 1-4, the graph of a function f is given. Tell whether f appears to be periodic. If so, give its

fundamental period and its amplitude, and then find ( )1000f and ( )1000−f .

5. Use the graph of ( )xfy = , shown at the right, to sketch the graph

of each of the following.

a. ( )xfy 2= b. ( )xfy2

1−= c. ( )xfy 2−=

d.

= xfy

2

1 e.

−=

2

1xfy f. ( ) 1+−= xfy


- 9 -

6. The greatest integer function xy = gives the greatest integer less

than or equal to x . Thus 21.2 = and 41.3 −=− . Use the graph

of this function, shown at the right, to sketch the graph of each of the following.

a. xy2

1= b. xy 2−= c.

−= xy

2

1

d. xy 2= e. 1−= xy f. 12 += xy

7. Sketch the graph of each of the following.

a. xy =+ 2 b. 3−= xy c. 54 +=− xy

d. 12 += xy e. xy −=+1 f. xy 23 =−

8. Sketch the graph of each of the following.

a. xy =−1 b. 4+= xy c. 52 −=+ xy

d. 32 −= xy e. xy −=− 2 f. xy 44 =−

9. Use the graph of xy 2= , shown at the right, to sketch the graph of each of

the following.

a. xy −= 2 b. 12 −= xy

c. xy 23 −= d. yx 2=

10. Use the graph of x

y1

= , shown at the right above, to sketch the graph of

each of the following.

a. x

y1

−= b. 2

1

−=

xy

c. x

y1

1+= d. y

x1

=

11. Given that the equation of a circle with radius 3 and center ( )0,0 is

922 =+ yx (graph at right), deduce the equation of the circle if it is translated so

that its center is ( )4,8 .

12. Refer to the circle with equation 922 =+ yx in Exercise 11. Sketch the

graph of 92

2

2

=+

y

x.


- 10 -

13. The graph of ( )xfy = , shown at the right, has x-intercepts at 0 and 6 and a

local maximum at (4, 32). a. Where do the x-intercepts and local maximum occur on the graph of

( )xfy 2= ?

b. Where do the x-intercepts and local maximum occur on the graph of

( )xfy 2= ?

c. Where do the x-intercepts and local maximum occur on the graph of

( )2−= xfy ?

d. Where do the x-intercepts and local maximum occur on the graph of

( )2+= xfy ?

e. If f is a cubic polynomial, find a rule for ( )xf .

14. a. Sketch the graph of xxxy 23 23 +−= and label all intercepts.

b. Sketch the graph of

+

−

= xxxy

2

12

2

13

2

123

by using the graph of part (a). Label all intercepts.

The first figure in Exercise 15 shows the calculator-drawn graph of xy = . Changes in the equation xy =

cause its graph to be reflected, stretched, shrunk, and/or translated to produce the graphs shown in parts (a)-(e). Give an equation for each graph. You may wish to use a graphing calculator to confirm your answers.

16. In this exercise, you will show that the graph of dcxbxaxy +++= 23 has a

point of symmetry at a

bx

3

−= .

a. If ( ) pxaxxfy +== 3 , show that ( ) ( )xfxf −=− .

b. Explain how part (a) shows that the origin is a symmetry point of the graph of

pxaxy += 3 .

c. Explain why ( )q,0 is a symmetry point for the graph of qpxaxy ++= 3 .

d. Explain why ( )qh, is a symmetry point for the graph of

( ) ( ) qhxphxay +−+−=3

.

e. Suppose the equation dcxbxaxy +++= 23 is rewritten in the equivalent form

( ) ( ) qhxphxay +−+−=3

. By comparing the coefficients of the 2x terms, show

that a

bh

3−= . Then use the results of part (d) to conclude that

dcxbxaxy +++= 23 has a point of symmetry at a

bx

3−= .


- 11 -

Practice Set F (pg 149)

1. Suppose a function f has an inverse. If ( ) 62 =f and ( ) 73 =f , find:

a. ( )61−f b. ( )( )31 ff − c. ( )( )71−ff

2. Suppose a function f has an inverse. If ( ) 10 −=f and ( ) 21 =−f , find:

a. ( )11 −−f b. ( )( )01 ff − c. ( )( )21−ff

3. If ( ) 53 =g and ( ) 51 =−g , explain why g has no inverse.

4. Explain why ( ) 23 xxxf += has no inverse.

5. Let ( ) 34 −= xxh .

a. Sketch the graphs of h and 1−h . b. Find a rule for ( )xh 1−

6. Let ( ) 42

1−= xxL .

a. Sketch the graphs of L and 1−L b. Find a rule for ( )xL 1− .

In Exercises 7-10, the graph of a function is given. State whether the function has an inverse.

State whether the function f has an inverse. If 1−f exists, find a rule for ( )xf 1− and show that

( )( ) ( )( ) xxffxff == −− 11 .

11. ( ) 53 −= xxf 12. ( ) 2−= xxf 13. ( ) 4 xxf = ; 0x ≥

14. ( )x

xf1

= 15. ( )2

1

xxf = 16. ( ) xxf −= 5 ; 5x ≤

17. ( ) 24 xxf −= 18. ( ) 25 xxf −= 19. ( ) 3 31 xxf +=

Sketch the graphs of g and 1−g . Then find a rule for ( )xg 1− .

20. ( ) 0 ,22 ≥+= xxxg 21. ( ) 0 ,9 2 ≤−= xxxg

22. ( ) ( ) 1 ,112

≤+−= xxxg 23. ( ) ( ) 4 ,142

≥−−= xxxg

Practice Set G (pg 161)

1. Express the area A of a °−°−° 906030 triangle as a function of the length h of the hypotenuse.

2. Express the area A of an equilateral triangle as a function of the perimeter P.

3. A tourist walks n km at 4 km/h and then travels 2n km at 36 km/h by bus. Express the total traveling time t (in hours) as a function of n.

4. A student holds a ball of string attached to a kite, as shown at the right. The string is held 1 m above the ground and rises at a °30 angle to the horizontal. If the student lets the string out at a rate of 2 m/s, express the kite’s height h (in meters) as a function of the time t (in seconds) after the kite begins to fly.

5. A store owner bought n dozen toy boats at a cost of $3.00 per dozen, and sold them at $.75 apiece. Express the profit P (in dollars) as a function of n.

6. The cost of renting a large boat is 30 dollars per hour plus a usage fee roughly equivalent to 3x cents per hour

when the boat is operated at a speed of x km/h. Express the cost C (in cents per kilometer) as a function of x.


- 12 -

7. The height of a cylinder is twice the diameter. Express the total surface area A as a function of the height h.

8. A pile of sand is in the shape of a cone with a diameter that is twice the height. Express the volume V of sand as a function of the height h.

9. A light 3 m above the ground causes a boy 1.8 m tall to cast a shadow s meters long measured along the ground, as shown at the right. Express s as a function of d, the boy’s distance in meters from the light.

10. When a girl 1.75 m tall stands between a wall and a light on the ground 15 m away, she casts a shadow h meters high on the wall, as shown at the right. Express h as a function of d, the girl’s distance in meters from the light.

11. A box with a square base has surface area (including the top) of 3 2m . Express the volume V of the box as a function of the width w of the base.

12. A box with a square base and no top has a volume of 6 3m . Express the total surface area A of the box as a function of the width w of the base.

13. A stone is thrown into a lake, and t seconds after the splash the diameter of the circle of ripples is t meters. a. Express the circumference C of this circle as a function of t. b. Express the area A of this circle as a function of t.

14. A balloon is inflated in such a way that its volume increases at a rate of 20 s/cm3 .

a. If the volume of the balloon was 100 3cm when the process of inflation began, what will the volume be after t seconds of inflation? b. Assuming that the balloon is spherical while it is being inflated, express the radius r of the balloon as a function of t.

Part (b) of Exercise 15 and 16 requires the use of a computer or graphing calculator.

15. Manufacturing A box with a square base and no top has volume 8 3m . The material for the base costs $8 per square meter, and the material for the sides costs $6 per square meter. a. Express the cost C of the materials used to make the box as a function of the width w of the base. b. Use a computer or graphing calculator to find the minimum cost.

16. Manufacturing A cylindrical can has a volume of 3cm 400π . The material for the top and bottom costs 2¢ per square centimeter. The material for the vertical surface costs 1¢ per square centimeter. a. Express the cost C of the materials used to make the can as a function of the radius r. b. Use a computer or graphing calculator to find the minimum cost.

17. At 2:00 p.m. bike A is 4 km north of point C and traveling south at 16 km/h. At the same time, bike B is 2 km east of C and traveling east at 12 km/h.

a. Show that t hours after 2:00 p.m. the distance between the bikes is 2080400 2 +− tt

b. At what time is the distance between the bikes the least? c. What is the distance between the bikes when they are closest?

18. A car leaves Oak Corners at 11:33 a.m. traveling south at 70 km/h. At the same time, another car is 65 km west of Oak Corners traveling east at 90 km/h. a. Express the distance d between the cars as a function of the time t after the first car left Oak Corners. b. Show that the cars are closest to each other at noon.


- 13 -

19. Engineering Water is flowing at a rate of 5 s/m3 into a conical tank shown at the right. a. Find the volume V of the water as a function of the water level h. b. Find h as a function of the time t during which water has been flowing into the tank.

20. Engineering A trough is 2 m long, and its ends are triangles with sides of length 1 m, 1 m, and 1.2 m as shown at the right. a. Find the volume V of the water in the trough as a function of the water level h. b. If water is pumped into the empty trough at the rate of 6 L/min, find the water level

h as a function of the time t after the pumping begins. ( L 1000m 1 3 = )

21. ( )yxP , is an arbitrary point on the line 102 =+ yx

a. Express the distance d from the origin to P as a function of the x-coordinate of P. b. What are the domain and range of this function?

22. ( )yxP , is an arbitrary point on the parabola 2xy = .

a. Express the distance d from P to the point A(0,1) as a function of the y-coordinate of P. b. What is the minimum distance d?

23. As shown at the right, rectangle ABCD has vertices C and D on the x-axis and

vertices A and B on the part of the parabola 29 xy −= that is above the x-axis.

a. Express the perimeter P of the rectangle as a function of the x-coordinate of A. b. What is the domain of the perimeter function? c. For what value of x is the perimeter a maximum?

24. Manufacturing A sheet of metal is 60 cm wide and 10 m long. It is bent along its

width to form a gutter with a cross section that is an isosceles trapezoid with °120 angles, as shown at the right. a. Express the volume V of the gutter as a function of x, the length in centimeters of one of the equal sides. (Hint: Volume = area of trapezoid × length of gutter) b. For what value of x is the volume of the gutter a maximum?

Part (b) of exercises 25-27 requires the use of a computer or graphing calculator.

25. From a raft 50 m offshore, a lifeguard wants to swim to shore and run to a snack bar 100 m down the beach, as shown. a. If the lifeguard swims at 1 m/s and runs at 3 m/s, express the total swimming and running time t as a function of the distance x shown in the diagram. b. Use a computer or graphing calculator to find the minimum time.

26. Engineering A power station and factory are on opposite sides of a river 60 m wide, as shown at the right above. A cable must be run from the power station to the factory. It costs $25 per meter to run the cable in the river and $20 per meter on land. a. Express the total cost C as a function of x, the distance downstream from the power station to the point where the cable touches the land. b. Use a computer or graphing calculator to find the minimum cost.


- 14 -

27. Landscaping A rectangular area of 60 2m has a wall as one of its sides, as shown. The sides perpendicular to the wall are made of fencing that costs $6 per meter. The side parallel to the wall is made of decorative fencing that costs $8 per meter. a. Express the total cost C of the fencing as a function of the length x of a side perpendicular to the wall. b. Use a computer or graphing calculator to find the minimum cost.

Practice Set H (pg 166)

1. Give the domain, range, and zeros fo the function ( ) 29 xxf −= .

2. Graph ( )

−≥−

−<+=

1 if 1

1 if 12

xx

xxxg . Find the range and zeros of g.

3. Let ( ) xxxf 22 += and ( ) 2+= xxg . Find:

a. ( )( )xgf + b. ( )( )xgf − c. ( )( )xgf ⋅ d. ( )xg

f

4. Using the functions f and g in Exercise 3, find:

a. ( )( )xgf � b. ( )( )xfg �

5. Determine whether the graph of 42 =− xyx has symmetry in: (i) the x-axis, (ii) the y-axis, (iii) the line

xy = , and (iv) the origin.

6. Given the graph of ( )xfy = shown, sketch the graph of each of the following.

a. ( )xfy 2= b. ( )xfy =

c. ( )xfy −= d. ( )2+= xfy

7. The graph of a periodic function ( )xgy = is shown.

a. What is the fundamental period of g? b. What are the maximum and minimum values of g? c. What is the amplitude of g?

8. Describe and illustrate what happens when the graph of a periodic function with period p is horizontally translated p units.

9. a. Which one of the two functions ( ) 23 xxf += and ( ) xxg += 3 has an inverse? Find a rule for the

inverse. b. Explain why the other function does not have an inverse.

10. The area A of a triangle is a function of the base b and height h. a. Express A as a function of b and h. b. Find A(3,4) and A(6,5).

c. Draw a curve of constant area 3),( =hbA in a bh-plane.

11. A cylindrical tank 4 ft in diameter fills with water at the rate of 10 s/ft3 . Express the depth of the water in the tank as a function of the time t in seconds. Assume the tank is empty at time t=0.


- 15 -

12. As shown, tirangle OAB is an isosceles triangle with vertex O at the origin and

vertices A and B on the part of the parabola 29 xy −= that is above the x-axis.

a. Express the area of the triangle as a function of the x-coordinate of A. b. What is the domain of the area function? c. Use a computer or graphing calculator to find the maximum area.

ANSWERS Practice Set A 1. Yes; D:

{ }52 ≤≤− xx R:

{ }21 ≤≤− yy

2.No; fails the vertical line test.

3. Yes; D: {0,2,4,6} R: {0,1,2,3}

4. Yes; D:

{ }2 6u u− ≤ ≤

R: { }1 3v v− ≤ <

5. No; fails the vertical line test.

6. Yes D: { }t t−∞ ≤ ≤ ∞ R:

{ }2 2s s− ≤ ≤

7. Not a fn, 21 xy −±= 8. Fn, 3 31 xy −=

9a. D: all real numbers; 0≠x 9b. D: all real numbers; 9≠x 9c. D:all real numbers; 2,2 −≠≠ xx

10a. D: all real numbers, 3t ≠ − 10b. D: all real numbers; 3, 2t ≠ − − 10c. D: all real numbers; 0, 3t ≠ ±

11a. D: all reals;

R: ( ) 0f x ≥ ; Zeros: 0=x

11b. D: all reals;

R: ( ) 0g x ≥ ; Zeros: 2=x

11c. D: all reals;

R: ( ) 2h x ≥ ; Zeros: 2±=x

12a. D: 0≥t ; R: ( ) 0f t ≥ ; Zeros:

0=t

12b. D: 9≤t ; R: ( ) 0g t ≥ ; Zeros:

9=t

12c. D: 33 ≤≤− t

R: 0 ( ) 3h t≤ ≤ ; Zeros: 3±=t

13.

R: ( ) 1f x ≥ −

Zeros: 2, 4

14.

R: ( ) 4g x ≤

Zeros: 1, 5

15.

R: all reals Zeros: 2

16.

R: all reals Zeros: -1, 1, -4

17.

R: 2 ( ) 4h u− < ≤

Zeros: 0, 2

18.

R: ( ) 0g u ≤

Zeros: 3

19a. 36π 19b. D: All solids; R: nonnegative real numbers

20a. S=sum of the measures of the interior angles of an n-gon

20b. Yes; D: inegers greater tan 2; R: positive multiples of 180


- 16 -

21a.

21b. D: all reals R: integers

22a.

22b.

( )3 if 0 1

3 2 if 1

tc t

t t

≤ ≤=

+ >

24a. y x= ± 24b. 21y x= ± − 24c. y x= ± 25. d 26. a, b

Answers Practice Set B

3a. 1

3b. pos: 40 << x ; neg: 01 <≤− x or

64 ≤< x ; zero: 0, 4

1.

2.

. 3c. 2

4a. -2 4b. pos: 52 ;01 <<<<− xx ; neg:

65 ;20 <<<< xx ; zero: 0, 2, 5

4c. 2

5. 23−+ xx 6. xx −

3 7. ( )( )113 −− xx 8. 12++ xx

9a. 0 9b. ( ) 113

−−x 10a. 6 10b. 23−x

13. zeros of f-g 14. 1.32 15. .64 16. 7071.± 11.

19a. 3 19b. 32 −x 20a. 3 20b. 12 −x

21a. 0 21b.

3

36 −x

22a. 3 22b. 32 −x 23. ( )( )( )xhfj

24. ( )( )( )xhfg

25.

( )( )( )xhjf 26. ( )( )( )xfhg 27. speed 1.5 m/s

Oxygen 10 L/min 28a. 4¢/mi 28b. bet 25-55 mph

29a. π2

Cr = 29b.

π4

2C

30a. 21; 2

2A r P r rπ π= = + 30b.

21

2 2

PA π

π

=

+

31. ( )

−+= 32

9

56.331 Fs 32a. ( )

2

Ar A

π= 32b. ( )

6

A AV A

π=

33. ( )( ) 2162 xxgf −=� D: 44 ≤≤− x

( )( ) 2416 xxfg −=� D: 22 ≤≤− x

34. ( )( )4

1

−=

xxgf � D: 4>x

( )( )4

1

−=

xxfg � D: 16or 16 >−< xx

35. ( )( ) xxgf −= 1� D: 1<x

( )( ) 21 xxfg −=� D: 10 ≤≤ x

36. ( )( ) 216 xxgf −=� D: 44 ≤≤− x

( )( ) 416 xxfg −=� D: 44 1616 ≤≤− x


- 17 -

37. 2.5, 2.75, 2.875 38. 8 7x −

Answers Practice Set C

1a.

1b.

1c.

2a.

2b.

2c.

3a.

3b.

3c.

4a.

4b.

4c.

5.


- 18 -

6.

7.

3

4

3

1+= xy

8.

22 −= xy

9.

yyx 22 −=

10.

yyx 32 +=

11.

3 xy =

12.

yx =

13.

2+= yx

14.

3−= yx

15a. no, no, no, yes 15b. yes all 15c. no, no, no, yes 16a. no, no, no, yes

16b. yes to all 16c. no, no, no, yes 21a. (3,5); 3x = 21b. (5,3); 3y =

22a. ( 1,3); 1x− = − 22b. (3, 1); 1y− = − 23a. (4,3); 4x = 23b. (3,4); 4y =

24. (4, 1); 1y− = − 25. ( 1, 3); 3y− − = − 26. ( 4, 1); 1y− − = −

27. max (-5, 11) 28a. (5, 55) 28b. (8,109)

29a. min (0,0) 29b. (1, 2); max (2, 4) 30a. min (-3, 0) 30b. (-2,2); max (-1, 4)

31a. even 31b. odd 31c. neither 31d. even 31e. neither 31f. odd

32. A polynomial function must contain only even powers of the variable to be and even function and only odd powers of the variable to be an odd function.


- 19 -

33a. In the y-axis 33b. In the origin 34. #3 is odd; #4 is even

Answers Practice Set D 7. B 8. E 9. H 10. D 11. I 12. A

13. L 14. C 15. F 16. J 17. G 18. K

19. ( )34−= xy 20. ( )3

4+= xy 21. 43 += xy 22. 43 −= xy

23. ( )3

y x= − 24. 3xy −= 25. 34xy = 26.

3

4

1

= xy

27. ( )

( )

2

2

2

y x

y x

y x

= +

= − +

= − − +

28. ( )

( ) 23

3

−−−=

−−=

−=

xy

xy

xy

29.

( ) 23

2

++−=

+−=

−=

xy

xy

xy

30.

( ) 23

2

2

++−=

+−=

+=

xy

xy

xy

31. C 32. D 33. C 34. C

35. a. -7 and 1 b. -3 and 5 c. -5 and 3 d. 5 and -3

36. a. -12 and -3 b. -5 and 4 c. -8 and 1 d. 8 and -1

39.

: ; : ( ) 1D R f xℜ ≥ −

40.

: ; : ( ) 4D R f xℜ ≥

41.

ℜℜ :;: RD

42.

ℜℜ :;: RD

43.

: 2; : ( ) 0D x R h x≥ ≥

44.

: 1; : ( ) 0D x R h x≥ − ≥

45.

ℜℜ :;: RD

46.

ℜℜ :;: RD

47.

: 0; : ( ) 0D x R g x≥ ≥

48.

: 0; : ( ) 0D x R g x≥ ≥

49.

}0{\:};0{\: ℜℜ RD

50.

ℜℜ :;: RD


- 20 -

51.

ℜℜ :;: RD

52.

: 0; : ( ) 0D x R f x≥ ≤

53.

ℜℜ :;: RD

54.

}0{\:};0{\: ℜℜ RD

55.

ℜℜ :;: RD

56.

}2{\:};0{\: ℜℜ RD

57.

: ; : ( ) 3D R f xℜ ≥ −

58.

: ; : ( ) 1D R f xℜ ≥

59.

: 2; : ( ) 1D x R g x≥ ≥

60.

: ; : ( ) 3D R g xℜ ≥ −

61.

: 0; : ( ) 2D x R h x≤ ≥ −

62.

}2{\:};0{\: ℜℜ RD

63.

ℜℜ :;: RD

64.

: 1; : ( ) 0D x R f x≥ ≤

65.

0: ;: ≥ℜ yRD

66.

: 2; : ( ) 0D x R g x≤ ≥


- 21 -

69a.

69b.

69c.

69d.

69e.

69f.

69g.

70a.

70b.

70c.

70d.

70e.

70f.

70g.

71a.

71b.

71c.

71d.


- 22 -

71e.

71f.

71g.

72a.

72b.

72c.

72d.

72e.

72f.

72g.

ANSWERS Practice Set E 1. per = 6; amp = 1

( ) 11000 −=f ; ( ) 11000 =−f

2. per = .8; amp = .5

( ) 01000 =f ; ( ) 01000 =−f

3. per = 3; amp = .5

( ) 21000 =f ; ( ) 31000 =−f

4. not periodic 5a.

5b.

5c.

5d.

5e.

5f.

6a.


- 23 -

6b.

6c.

6d.

6e.

6f.

7a.

7b.

7c.

7d.

7e.

7f.

8a.

8b.

8c.

8d.

8e.

8f.

9a.

9b.

9c.

9d.

10a.

10b.

10c.

10d.

11. ( ) ( ) 94822

=−+− yx 12.

13a. x-int: 0, 3; max: (2, 32) 13b. x-int: 0, 6; max: (4, 64) 13c. x-int: 2, 8; max: (6, 32)

13d. x-int: -2, 4; max: (2, 32) 13e. ( ) ( )62 −−= xxxf 15a. 42 −=− xy

15b. xy −=− 4 15c. xy

2

12 =+

15d. 2y x= − 15e. 42 += xy

ANSWERS Practice Set F 1a. 2 1b. 3 1c. 7 2a. 0 2b. 0 2c. 2 3. not one-to-one 4. not one-to-one


- 24 -

5a.

5b. ( ) ( )34

11 +=−xxh

6a.

6b. ( ) ( )421 +=− xxL

7. inverse 8. no inverse 9. no inverse 10. inverse

11. ( ) ( )53

11 +=−xxf

12. no inverse 13. ( ) 0 ;41 ≥=− xxxf 14. ( )

xxf

11 =−

15. no inverse 16. ( ) 0 ;5 21 ≥−=− xxxf 17. no inverse 18. no inverse

19. ( ) 3 31 1−=− xxf

20.

( )1 2; 2g x x x− = − ≥

21.

( )1 9 ; 9g x x x− = − − ≤

22.

( )1 1 1; 1g x x x− = − − + ≥

23.

( )1 1 4; 1g x x x− = + + ≥ −

ANSWERS Practice Set G

1. 8

32hA = 2.

36

32pA = 3.

36

2

4

nnT +=

4. ( ) 11 += tth

5. ( ) nnP 31275. −= 6. 33000 xC += 7.

25

8

hSA

π= 8. 3

3

1hV π=

9. ds2

3= 10.

dh

25.26= 11.

( )4

23 2ww

V−

= 12. w

wSA242 +=

13a. C tπ= 13b.

2

2

tA π

=

14a. tV 20100 += 14b. 3

1575

π

tr

+=

15a. 2 1928C w

w= +

15b. $125.80 16a.

rrC

ππ

804. 2 +=

16b. $8.12

17b. .1 hr 17c. 4 km 18a. ( ) ( )22

709065 ttd +−=

19a. 48

3h

Vπ

= 19b. 3240

π

th =

20a. 2723. hV = 20b. th 0083.=

h

1h

− L

1L−

1g

−

g

g

1g

−

g

1g

−


- 25 -

21a.

100405 2 +−= xxd

21b. D: x ∈ℜ

R: ( ) 20d x ≥ 22a. 124 +−= xxd

22b. 4

21

23a. 22418 xxP −+= 23b. 33 ≤≤− x 23c. 20

24a. ( )xxV 31203250 −= 24b. 519615 3cm 25a.

3

100

1

5022xx

t−

++

=

25b. 80.47 seconds 26a. ( )xxC −++= 200206025 22 26b. $4900

27a. ( )

+=

xxC

60826

27b. $151.79

Practice Set H

1. domain: 33 ≤≤− x ; range: 0 ( ) 3f x≤ ≤ ; zeros: -3, 3

2. graph; range: ( ) 1g x ≤ ; zeros: -

1, 1 3a. ( )( ) 232 ++=+ xxxgf

3b. ( )( ) 22 −+=− xxxgf 3c. ( )( ) xxxxgf 44 23 ++=⋅ 3d. ( ) ; 2

fx x x

g

= ≠ −

4a. ( )( ) 862 ++= xxxgf � 4b. ( )( ) ( ) 222 ++= xxxfg � 5. no, no, no, yes

6. graphs 7. per = 4, max = 4, min = 1, amp = 1.5

9a. ( )xf no inverse

( ) 31 −=− xxg

10a. hbA ⋅=2

1

10b. ( ) 64,3 =A ; ( ) 155,6 =A 11.

π4

10td =

12a. ( )29 xxA −= 12b. 33 <<− x 12c. max 10.39


- 26 -

Practice Set A Graphs (use for homework)

Practice Set B Graphs (to use for homework)

1.

3.

4.

Practice Set C Graphs (to use for homework)

1a.

1b.

1c.

2a.

2b.

2c.


- 27 -

3a.

3b.

3c.

4a.

4b.

4c.

Practice Set D Graphs (to use for homework)

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

69a.

69b.

69c.

69d.


- 28 -

69e.

69f.

69g.

70a.

70b.

70c.

70d.

70e.

70f.

70g.

71a.

71b.

71c.

71d.

71e.

71f.

71g.

72a.

72b.

72c.

72d.


- 29 -

72e.

72f.

72g.

Practice Set E Graphs (to use for homework)

5a. 5b. 5c.

5d. 5e. 5f.

6a.

6b.

6c.

6d.

6e.

6f.


- 30 -

9a.

9b.

9c.

9d.

10a.

10b.

10c.

10d.

Practice Set H Graphs (to use for homework)

6a.

.

6b.

6c.

6d.


- 31 -


- 32 -

LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME … A… · 5. Sketch the graphs of y = x2 − 9, y = 9− x2 , and y = 9− x2 on a single set of axes. 6. Sketch the graphs of

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