Toolkit of Functions

Students should know the basic shape of these functions and be able to

graph their transformations without the assistance of a calculator.

Constant Cubic

f(x) = a f(x) = x3

Identity Square Root

f(x) = x f(x) = x

Absolute Value Greatest Integer

f(x) = | x | f(x) = [ x ]

Reciprocal Exponential

f(x) = 1

x f(x) =

xa

Quadratic Logarithmic

f(x) = x2 f(x) = ln x

Trig Functions

( ) sin ( ) cos ( ) tanf x x f x x f x x

Polynomial Functions: A function P is called a polynomial if

�� � � � � �1 2

1 2 1( ) ...

n n

n nP x a x a x a x a x a

0

Where n is a nonnegative integer and the numbers a0, a1, a2, … an are constants.

Even degree Odd degree

Leading coefficient sign Leading coefficient sign

Positive Negative Positive Negative

x Number of roots equals the degree of the polynomial.

x Number of x intercepts is less than or equal to the degree.

x Number of “bends” is less than or equal to (degree – 1).

Formulas and Identities

Trig Formulas:

Arc Length of a circle: L = rT or L = d

r360

2x S

Area of a sector of a circle: Area = 212

r T or Area = d

r360

2x S

Solving parts of a triangle:

Law of Sines: a

Ab

Bc

Csin sin sin

Law of Cosines: a b c bc A2 2 2 2 � � cos

b a c ac2 2 2 2 B � � cos c a b ab C2 2 2 2 � � cos Area of a Triangle:

Area = 12

bc sinA or Area = 12

ac sinB or Area = 12

ab sinC

Hero's formula : Area = s s a s b s c( )( )(� � � ) , where s = semi perimeter Ambiguous Case: is acute T T is obtuse or right Compute alt adj: s x Tin opp d adj No triangle opp < alt No triangle opp > adj 1 triangle opp = alt 1 triangle (right) opp>adj 1 triangle alt<opp<adj 2 triangles Does a triangle exist? Yes - when (difference of 2 sides) < (third side) < (Sum of 2 sides)

Formulas and Identities, continued

Trig Identities:

Reciprocal Identities:

csc A = 1

sinA sec

cosA

A

1 cot A =

1tanA

Quotient Identities: tansincos

AAA

cot A =cossin

AA

Pythagorean Identities:

sin2A + cos2A = 1 tan2A + 1 = sec2A 1 + cot2A = csc2A Sum and Difference Identities:

sin(A + B) = sinA cosB+cosA sinB sin(A – B) = sinA cosB - cosA sinB

cos(A + B) = cosA cosB – sinA sinB cos(A – B) = cosA cosB + sinA sinB

tan (A + B) = tan tan

tan tanA B

A B�

�1 tan (A – B) =

tan tantan tanA B

A B�

�1

Double Angle Identities:

sin(2A) = 2sinA cosA tan(2A) = 2

1 2tantan

AA�

cos(2A) = cos2A - sin2A cos(2A) = 2cos2A – 1 cos(2A) = 1 – 2sin2A Half Angle Identities:

1 cossin

2 2A A� r

1 costan

2 1 cosA A

A�

r�

1 coscos

2 2A A� r

Polar Formulas:

2 2 2x y r� cos sinx r y r T T 1 1tan 0, tan 0y yx xx x

� �T ! T� S �

GeometricȱFormulas:ȱȱȱȱȱȱAreaȱofȱaȱtrapezoid:ȱȱ � 1 2

12

A h b b � � ȱȱȱȱȱȱȱȱȱAreaȱofȱaȱtriangle:ȱȱ 12

A b h ȱ

ȱȱȱȱȱAreaȱofȱanȱequilateralȱtriangle:ȱ 234

A s ȱȱȱȱȱȱ

ȱȱȱȱȱAreaȱifȱaȱcircle:ȱȱ 2A rS ȱȱȱȱȱCircumferenceȱofȱaȱcircle:ȱC r2 or C dS S ȱȱȱ

Unit Circle – Degrees and Radians ȱ

Place degree measures in the circles. tanT = ________________

Place radian measure in the squares. cotT= _________________

Place (cos , sin ) in parenthesis outside the square. T T cscT = ________________

Place tan outside the parenthesis. T secT= ________________

SKILLS NEEDED FOR CALCULUS

I. Algebra: *A. Exponents (operations with integer, fractional, and negative exponents) *B. Factoring (GCF, trinomials, difference of squares and cubes, sum of cubes, grouping) C. Rationalizing (numerator and denominator) *D. Simplifying rational expressions *E. Solving algebraic equations and inequalities (linear, quadratic, higher order using synthetic division, rational, radical, and absolute value equations) F. Simultaneous equations

II. Graphing and Functions *A. Lines (intercepts, slopes, write equations using point-slope and slope intercept, parallel, perpendicular, distance and midpoint formulas) B. Conic Sections (circle, parabola, ellipse, and hyperbola) *C. Functions (definition, notation, domain, range, inverse, composition) *D. Basic shapes and transformations of the following functions (absolute value, rational, root, higher order curves, log, ln, exponential, trigonometric. piece-wise, inverse functions) E. Tests for symmetry: odd, even

III. Geometry

A. Pythagorean Theorem B. Area Formulas (Circle, polygons, surface area of solids) C. Volume formulas D. Similar Triangles

* IV. Logarithmic and Exponential Functions

*A. Simplify Expressions (Use laws of logarithms and exponents) *B. Solve exponential and logarithmic equations (include ln as well as log) *C. Sketch graphs *D. Inverses

* V. Trigonometry

**A. Unit Circle (definition of functions, angles in radians and degrees) B. Use of Pythagorean Identities and formulas to simplify expressions and prove identities *C. Solve equations *D. Inverse Trigonometric functions E. Right triangle trigonometry *F. Graphs

VI. Limits

A. Concept of a limit B. Find limits as x approaches a number and as x approaches � * A solid working foundation in these areas is very important.

Calculus Prerequisite Problems

Work the following problems on your own paper. Show all necessary work.

I. Algebra

A. Exponents: 1) � � � �

� �1 33 3

11 23 3

8 2

4

x yz x

x yz�

B. Factor Completely:

Hint: Factor as difference of squares first, then as the sum and difference of cubes second. 2) 9x2 + 3x - 3xy - y (use grouping) 3) 64x 6 - 1

4) 42x 4 + 35x 2 - 28 5) 5 3 1

2 2 22 24� �15 x x x Hint: Factor GCF x 1/2 first.

6) x -1 -3x -2 + 2x -3 Hint: Factor out GCF x-3 first.

C. Rationalize denominator / numerator:

7) x

x�

� �3

1 2 8)

x + 1 + 1 x

D. Simplify the rational expression:

9) (x + 1)3 (x - 2) + 3(x + 1 )2

(x + 1)4

E. Solve algebraic equations and inequalities

10. – 11. Use synthetic division to help factor the following, state all factors and roots.

10) p(x) = x3 + 4x 2 + x - 6 11) p(x) = 6x3 - 17x 2 - 16x + 7

12) Explain why 32 cannot be a root of f(x) = 4x5 + cx3 - dx + 5 , where c and d are integers.

(hint: You can look at the possible rational roots.)

13) Explain why f(x) = x4 + 7x2 + x - 5 must have a root in the interval [0, 1], ( 0 � x � 1) Check the graph and use signs of f(0) and f(1) to justify you’re answer.

Solve: You may use your graphing calculator to check solutions.

14) (x + 3)2 > 4 15) x + 5x - 3 d 0 16) 3 23 14 5 0� � dx x x (Factor first)

17) x < 1x 18)

x2 - 9 x + 1 t 0 19)

1 x - 1 +

4 x - 6 > 0

20) x 2 < 4 21) | 2x + 1 | < 14

F. Solve the system. Solve the system algebraically and then check the solution by graphing each function and using your calculator to find the points of intersection.

22) x - y + 1 = 0 23) x 2 - 4x + 3 = y y – x 2 = - 5 - x 2 + 6x - 9 = y II. Graphing and Functions: A. Linear graphs: Write the equation of the line described below.

24) Passes through the point (2, -1) and has slope - 13 .

25) Passes through the point (4, - 3) and is perpendicular to 3x + 2y = 4.

26) Passes through ( -1, - 2) and is parallel to y = 35 x - 1.

B. Conic Sections: Write the equation in standard form and identify the conic.

27) x = 4y 2 + 8y - 3 28) 4x 2 - 16x + 3y 2 + 24y + 52 = 0 C. Functions: Find the domain and range of the following. Note: domain restrictions - denominator � 0, argument of a log or ln > 0, radicand of even index must be � 0 range restrictions- reasoning, if all else fails, use graphing calculator

29) y = 3

x - 2 30) y = log(x - 3) 31) y = x 4 + x2 + 2

32) y = 2x - 3 33) y = | x - 5 | 34) domain only: 2

11

�

�xy

x

35) Given f(x) below, graph over the domain [ -3, 3], what is the range?

if x 0

( ) 1 if -1 x < 02 if x < -1

xf x

x

t­° d®° �¯

Find the composition /inverses as indicated below.

Let f(x) = x2 + 3x - 2 g(x) = 4x - 3 h(x) = lnx w(x) = x - 4

36) g -1(x) 37) h -1(x) 38) w -1(x), for x � 4 39) f(g(x)) 40) h(g(f(1)))

41) Does y = 3x2 - 9 have an inverse function? Explain your answer. Let f(x) = 2x, g(x) = -x, and h(x) = 4, find

42) (f o g)(x) 43) (f o g o h)(x) 44) Let s (x) = 4 - x and t(x) = x2, find the domain and range of (s o t )(x). D. Basic Shapes of Curves:

Sketch the graphs. You may use your graphing calculator to verify your graph, but you should be able to graph the following by knowledge of the shape of the curve, by plotting a few points, and by your knowledge of transformations.

45) y = x 46) y = lnx 47) y = 1x 48) y = | x - 2 |

49) y = 1

x - 2 50) y = x

x2 - 4 51) y = 2-x 52) y = 3 sin 2 (x - Ǒ6 )

53)

2

2

25 if x < 0

25( ) if x 0, x 5

50 if x = 5

x

xf x

x

­ �°

�° t® �°°¯

z

E. Even, Odd, Tests for Symmetry:

Identify as odd, even , or neither and justify you’re answer. To justify your answer you must show substitution using -x ! It is not enough to simply check a number.

Even: f (x) = f (-x) Odd: f (-x) = - f (x)

54) f(x) = x3 + 3x 55) f(x) = x 4 - 6x2 + 3 56) 3

2( )

x xf x

x�

57) f(x) = sin 2x 58) f(x) = x2 + x 59) f(x) = x(x2 - 1)

60) f(x) = 1 + |x|

x2

61) What type of function (even or odd) results from the product of two

even functions? odd functions?

Test for symmetry. Show substitution with variables to justify you’re answer.

Symmetric to y axis: replace x with - x and relation remains the same. Symmetric to x axis: replace y with - y and relation remains the same. Origin symmetry: replace x with - x, y with - y and the relation is equivalent. 62) y = x 4 + x 2 63) y = sin(x) 64) y = cos(x)

65) x = y2 + 1 66) y = |x|

x2 + 1

IV LOGARITHMIC AND EXPONENTIAL FUNCTIONS A. Simplify Expressions:

� � � �

� �

) log ) log - log log ) log

70) log ) ln 73) l 71) log

4 3 3 3 9

45125

1 116 27

15

3

4

167 68 3 3 81 69 27

3

72w ew

�

n 74) ln 21 e

B. Solve equations:

75) log6(x + 3) + log6(x + 4) = 1 76) log x2 - log 100 = log 1 77) 3 x+1 = 15 V TRIGONOMETRY

A. Unit Circle: Know the unit circle – radian and degree measure. Be prepared for a quiz.

78) State the domain, range and fundamental period for each function? a) y = sin x b) y = cos x c) y = tan x

B. Identities:

Simplify: 79) (tan2x)(csc2x) - 1(cscx)(tan2x)(sinx) 80) 1 - cos2 x 81) sec2x - tan2 x

82) Verify : (1 - sin2x)(1 + tan2x) = 1 C. Solve the Equations

83) cos2x = cos x + 2, 0 � x � 2Ǒ 84) 2 sin(2x) = 3 , 0 � x � 2Ǒ 85) cos2x + sinx + 1 = 0, 0 � x � 2S

D. Inverse Trig Functions: Note: Sin -1 x = Arcsin x

2 386) Arcsin1 87) Arcsin 88) Arccos 89) sin Arccos

2 2

§ ·§ · § · § ·� ¨ ¸¨ ¸ ¨ ¸ ¨ ¸¨ ¸ ¨ ¸ ¨ ¸¨ ¸© ¹ © ¹ © ¹© ¹

32

90. State domain and range for: Arcsin(x) , Arccos (x), Arctan (x) E. Right Triangle Trig: Find the value of x. (Note: Degree measure!)

x 10

50o

70 70o o

X

10

93.90. 91.92

91.

180 ft 60 ftA B C

H

93) The roller coaster car shown in the diagram above takes 23.5 sec. to go up the 23 degree incline segment AH and only 2.8 seconds to go down the drop from H to C. The car covers horizontal distances of 180 feet on the incline and 60 feet on the drop. Decimals in answer may vary.

a. How high is the roller coaster above point B? b. Find the distances AH and HC. c. How fast (in ft/sec) does the car go up the incline?

d. What is the approximate average speed of the car as it goes down the drop? e. Assume the car travels along HC. Is your approximate answer too big or too small? ( Advanced Mathematics, Richard G. Brown, Houghton Mifflin,1994, pg 336)

F. Graphs: Identify the amplitude, period, horizontal, and vertical shifts of these functions.

94) y = - 2sin(2x) 95) � �cos 2SS S � �y x

G. Be able to do the following on your graphing calculator:

Be familiar with the CALC commands; value, root, minimum, maximum, intersect. You may

need to zoom in on areas of your graph to find the information. Answers should be accurate to 3 decimal places. Sketch graph. 96. – 99. Given the following function f(x) = 2x 4 - 11x 3 - x 2 + 30x. 96. Find all roots. Note: Window x min: -10 x max: 10 scale 1 y min: - 100 y max: 60 scale 10 97. Find all local maxima. A local maximum or local minimum is a point on

the graph where there is a highest or lowest point within an interval such as the vertex of a parabola.

98. Find all local minima. 99. Find the following values: f (-1), f (2), f (0), f (.125) 100. Graph the following two functions and find their points of intersection using the intersect command on your calculator. y = x3 + 5x2 - 7x + 2 and y = .2x2 + 10 Window: x min : -10 x max: 10 scale 1 y min: - 10 y max: 50 scale 0

VI. Functions and Models

101. The graphs of f and g are given.

(a) State the values of f(-4) and g(3). (b) For what values of x if f(x)=g(x)? (c) Estimate the solution of the equation f(x) = -1. (d) On what interval is f decreasing? (e) State the domain and range of f. (f) State the domain and range of g.

102. The number N (in thousands) of cellular phone subscribers in Malaysia is shown in the table. (Midyear estimates are given.)

t 1991 1993 1995 1997 N 132 304 873 2461

(a) Use the data to sketch a rough graph of N as a function of t. (b) Use your graph to estimate the number of cell-phone subscribers in Malaysia at midyear in 1994 and 1996.

103. If f(x) = 3x 2 -x+2, find f(2), f(-2), f(a), f(-a), f(a+1), 2f(a), f(a ), [f(a)] 2 , and f(a+h). 2

104. Find the domain of each function.

a) ( )3 1

xf xx

�

b) ( ) 4g u u u � � 105. Find an expression for the bottom half of the parabola x+(y-1) =0. 2

106. A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides. 107. Find an expression for the function whose graph is the given curve.

108. Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70° F and 173 chirps per minute at 80° F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. 109. At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lb/in . Below the surface, the water pressure increases by 4.34 lb/in for every 10 ft of descent.

2 2

(a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 100 lb/in ? 2

110. Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, or logarithmic function.

22 9 45

3

1( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )

xa f x x b g x x c h x x x d r xx x

� � �

�

8

111. Match each equation with its graph. Explain your choices. (Don’t use a computer or graphing calculator). (a) 2 5( ) ( )y x b y x c y x

112. Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f) Reflect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3. 113. The graph of y = f(x) is given. Match each equation with its graph and give reasons for your choices. (a) y = f(x-4) (b) y = f(x)+3 (c) y = 1/3 f(x) (d) y = -f(x+4) (e) y = 2f(x+6)

114. The graph of f is given. Use it to graph the following functions. (a) y = f(2x) (b)=f( ½ x) (c) y = f(-x) (d) y = -f(-x)

115. Graph the following, not by plotting points, but by starting with the graph of one of the standard functions and applying the appropriate transformations.

1sin

3 6y x S§ · �¨ ¸

© ¹

116. Find f+g, f – g, fg, and f/g and state their domains.

3 2 2( ) 2 , ( ) 3 1f x x x g x x � � 117. Find the functions , , , andf g g f f f g$ $ $ g$ and their domains.

( ) sin , ( ) 1f x x g x x� 118. Express the function in the form f g$ .

2 1( ) ( 1)F x x � 0 119. Use a graphing calculator to determine which of the given viewing rectangles produces the most appropriate graph of the function 3( ) 10 25f x x x � � . (a) [-4,4} by [-4,4] (b) [-10,10] by [-10,10] (c) [-20,20] by [-100, 100] (d) [-100, 100] by [-200,200]

120. Use the given graphs of f and g to evaluate each expression, or explain why it is undefined. (a) f(g(2)) (b) g(f(0)) (c) ( )(0) ( ) ( )(6) ( ) ( )( 2) ( ) ( )(4f g d g f e g g f f f )�$ $ $ $

121 and 122, Determine an appropriate viewing rectangle for the given function and use it to draw each graph. 121. 2( ) 5 20f x x � � x

122. 44( ) 81f x x � 123. Graph the ellipse by graphing the functions whose graphs are the upper and lower halves of the ellipse.

2 24 2x y� 1

0

124. Find all solutions of the equation correct to three decimal places.

3 29 4x x� � 125. Graph the given functions on a common screen. How are these graphs related?

2 , , 5 , 20x x xy y e y y x 126. Starting with the graph of y = e x , write the equation of the graph that results from (a) shifting 2 units downward (b) shifting 2 units to the right (c) reflecting about the x-axis (d) reflecting about the y-axis (e) reflecting about the x-axis and then about the y-axis

127. Find the exponential function f(x) = Ca x whose graph is given.

128. Under ideal conditions a certain bacteria population is known to double every three hours. Suppose that there are initially 100 bacteria. (a) What is the size of the population after 15 hours? (b) What is the size of the population after t hours. (c) Estimate the size of the population after 20 hours. (d) Graph the population function and estimate the time for the population to reach 50,000. 129. Determine whether this function is one-to-one.

x 1 2 3 4 5 6 f(x) 1.5 2.0 3.6 5.3 2.8 2.0 130. Determine whether this function is one-to-one.

131. Here is a verbal description of a function. Determine whether this function is one-to-one. F(t) is the height of a football t seconds after kickoff. For #132-134, find a formula for the inverse of the function. 132. ( ) 10 3f x x �

133. 3

( ) xf x e 134. y = ln(x+3)

For #135-136, find the exact value of each expression (non-calculator).

135. (a) 2 61

log 64 ( ) log36

b

136. (a) 10 10 5 5 5log 1.25 log 80 ( ) log 10 log 20 3log 2b� � � 137. Express the given quantity as a single logarithm. 2 ln 4 – ln 2 138. The graph shown gives a salesman’s distance from his home as a function of time on a certain day. Describe in words what the graph indicates about his travels on this day.

Answers: (Remember – you must show all of your work!)

. 4x11/3y4/3z 2. (3x + 1)(3x - y) 3. (2x - 1)(4x2 + 2x + 1)(2x + 1)(4x2 - 2x +1)

. 7(3x2 + 4)(2x2 - 1) 5. x1/2(3x - 4)(5x + 6)

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

1 4

9. (x + 1)2 x2 - x + 1

. 1 + x - 2 8. 1

71 1� �x

6. x

10. (x –1)(x + 2)(x + 3); 1, -2, -3 11. (x + 1)(2x – 7)(3x – 1); -1, 7 2 ,

1 3

nd f(1) = positive 14. x> -1 or x < -5 15. -5 d x < 3

12. not a possible rational

root 13. f(0) = neg a

. or 0 x 53

0 <

5

116 d � d dx 17. x < 1 or x < -1 18. [ -3, -1) U [3, �) 19. x > 6 or 1 < x < 2

20. –2 < x < 2 21. - 8 < x < - 8 3

22. (3, 4), (-2, -1) 23. (2, -1) , ( 3, 0)

24. 25. 26. 1 1 2 17 33 3 3

7 � � � �y x y x y x 27. x = 4(y + 1)2 – 7 parabola

3 5 5� � � �

.28 13 4

� ellipse 29. D: x � 2 R: y � 0 30. : 3 R:all realsD x ! 2 22 4� �x y

1. D: all reals R: x � 2 32. D: x > 3 3/2 R: y > 0 33. D: all reals R: y � 0

34. x >-1 and x � 1 35. R: - 5 � y < - 3 or 0 � y � 3 36. ( )g x 4

37. h -1(x) x� �1 3

= ex

8. y = x2 + 4 x >3 0, 39. f(g(x)) = 16x2 – 12x – 2 40. ln 5 41. no, not 1 to 1 42. –2x

3. - 8 44. D: R:

4 2 2x� d d 0 2yd d

45. 47. 49. 51. 53.

You must show work on these! 54. odd 55. even 56. odd 57. odd 58. neither 59. odd

60. even 61. even, even 62. symm y axis 63. origin 64. y axis 65. x axis 66. y axis

67. - 2 68. –1 69. 3/2 70. – 1/3 71. 45 72. 1 73. 0 74. 2 75. - 1

76. x = 10, -10 77.

log 15 log 3 - 1 78. a) D: all reals, R: -1 < x < 1 , 2 ș

) D: all reals, R: -1 < ,2

n where n ZS Sr � R: all reals , Ǒ 79. 1 80. sin2x b x < 1 , 2 ș c) D: x �

81. 1 82. yes 83. S 84. , , , . .S S S S S7 4 3

85 866 3 6 3 2 2

S

. .S S

�1

87 88 894 6

.2

90. Arcsin(x) D: [-1, 1] Range: ,2 2S Sª º�« »¬ ¼

,

Arc cos (x) D: [-1, 1]. R: [0, Ǒ] Arctan (x) D: all reals R: ,S S§ ·�¨ ¸

© ¹2 2

91. 10 sin 50 7.66 92. hint: draw altitude , x � � 14.619 93. a. HB � 76.405 ft.

. AH 195.5448 ft., HC 97.148 c. speed up approx. 8.32 ft./sec b � �

. speed down approx 34.696 ft./sec e. Too small, since the distance traveled is longer the d

distance per foot would be greater. (Note: find average speed for the entire trip! 11.129 ft/sec)

94. A: 2, P: Ǒ 95. A = S , Per. 4, v.s. = 0 , h. s. = -2 y = � �cos 22

xSS ª º� �« »¬ ¼ 96. - 1.5, 0 , 2 , 5

97. rel max. ( 1.07, 20.1) 98. rel. min ( - .89, - 18. 48), (3.94, - 88)

9. f( -1) = - 18 100. 3 points of intersection - one is ( -5.77, 16.66)

00. Do not use trace to find points. Use CALC commands.

104. (a) (-�, 1/3) U (1/3,�) (b) [0,4] 105.

9

1

,4] 101. (a) -2,4 (b) -2 and 2 (c) x= -3 and 4 (d) [0,4] (e) d: [-4,4] r:[-2,3] (f) d: [-4,3] r:[0.5102. (b) 540 In 1994 and 1450 In 1996 103. f(2)=12, f(-2) = 16, f(a) = 2a^2 -a +2,

2a+2, f(a^2)=3a^4 -a^2 +2, f(-a)=3a^2+a+2, f(a+1)=3a^2+5a+4, 2f(a)=6a^2-2a+4, f(2a)=12a^2-[f(a)]^2=9a^4 -6a^3+13a^2-4a+4, f(a+h)=3a^2 +6ah+3h^2 -a-h+2

( ) 1f x x � � (domain: x � 0)

106. A(L) = 10L –L^2 domain 5<L<10 107. 1 1 2

( ) 36 2 4

2°̄

x if xf x

x if x

� � d d­° ®� � � d

icket chirps

ecause it unction

the y-axis (d) reflect the graph of f about the y-axis, then about the x-axis 115.

108. (a) T=1/6 N + 307/6 (b) 1/6 means that for each increase of 6 cr per minute corresponds to an increase of 1 degree F. (c) 76 degrees F 109. (a) P=0.434d + 15 (b) approx. 196 feet 110. (a) root fct. (b) algebraic bIs the root of a polynomial (c) polynomial of degree 9 (d) rational f111. (a) matches with h (b) matches with f and (c) matches with g. 112. (a) y = f(x)+3 (b) y = f(x) -3 (c) y = f(x-3) (d) y = f(x+3) (e) y = -f(x) (f) y = f(-x) (g) y = 3f(x) (h) y = 1/3 f(x) 113. (a) graph 3 (b) graph 1 (c)graph 4 (d) graph 5 (e) graph 2 114. (a) shrink horizontally by a factor of 2 (b) stretch horizontally by a factor of 2 (c) reflect the graph of f about

116. (f+g)(x) =x^3+5x^2-1 d: all real numbers (f-g)(x)= x^3-x^2+1 d: all reals (fg)(x)=3x^5+6x^4-x^3-2s^2 d: all reals (f/g)(x)=(x^3 +2x^2)/(3x^2 -1) d: x cannot

Equal 13

r 117. ( )( ) sin(1 ) :[0, ) ( )( ) 1 sin :[0, ],[2 ,3 ]f g x x d g f x x d etcS S S � f �$ $

( )( ) sin(sin ) : ( , ) ( )( ) 1 1 :[0, )f f x x d g g x x d �f f � �$ $ f 118. g(x)=x^2 +1 and F(x)=x^10 119. c 120.(a) 4 (b) 3 (c)0 (d) not defined (e) 4 (f) -2 121. [-10,30] by [-50,150] 122. [-4,4] by [-1,4] 123. graph y = +sqr((1-4x^2)/2) and -sqr((1-4x^2)/2) 124. about 9.05 125. All of these graphs approach 0 as x approaches negative infinity, all of them pass Through the point (0,1) and all of them are increasing and approach infinity as x Approaches infinity. 126. (a) y = e^x -2 (b) y = e^(x-2) (c) y = -e^x (d) y = e^(-x) (e) y = - e^(-x) 127. f(x) = 3(2^x) 128. (a) 3200 (b) y = 100(2^(t/3)) (c) 10,159 (d) 26.9 hrs. 129. No, does not pass horizontal line test. 130. No 131. A football will attain every height up to its maximum height twice: once on the way up, and again on the way down.

Hence, not 1 to 1. 132. 1 21 1( )

3 3f x x� � �

0 d: [0,�) 133. 1 3( ) lnf x� x 134. 1( ) 3xf x e� �

135 (a) 6 (b) -2 136. (a) 2 (b) 2 137. ln8 138. The salesman travels away from home from 8 am To 9 am and is then stationary until 10:00. The salesman travels farther away from 10 until noon. There is no change in his distance from home until 1:00, at which time the distance from home decreases until 3:00. Then the distance starts increasing again, reaching the maximum distance away from home at 5:00. There is no chance from 5 until 6, and then the distance decreases rapidly until 7:00 pm, at which time the salesman reaches home.