unctions from a Iculus Perspective • In Algebra 2, you analyzed functions from a graphical perspective. _ O In Chapter 1, you will: • Explore symmetries of gra| • Determine continuity and average rates of change of functions. • Use limits to describe end behavior. • Find inverse functions algebraically and graphically. USINESS Functions are world. Some of the uses of functions are to analyze costs, predict sales, calculate profit, forecast future costs and revenue, estimate depreciation, and determine the proper labor force. PREREAD Create a list of two or three things that you already know about functions. Then make a prediction of what you will learn in Chapter 1. LconnectED.mcgraw-hill.com Animation Vocabulary Your Digital Math Porta eGlossary Personal Tutor Graphing Calculator Self-Check Practice Worksheets
Transcript
unctions from a Iculus Perspective
•
In Algebra 2, you analyzed functions from a graphical perspective.
_
O In Chapter 1, you will:
• Explore symmetries of gra|
• Determine continuity and average rates of change of functions.
• Use limits to describe end behavior.
• Find inverse functions algebraically and graphically.
USINESS Functions are world. Some of the uses of functions are to analyze costs, predict sales, calculate profit, forecast future costs and revenue, estimate depreciation, and determine the proper labor force.
PREREAD Create a list of two or three things that you already know about functions. Then make a prediction of what you will learn in Chapter 1.
LconnectED.mcgraw-hill.com
Animation Vocabulary
Your Digital Math Porta
eGlossary Personal Tutor
Graphing Calculator
Self-Check Practice Worksheets
Get Ready for the Chapter Diagnose Readiness You have two options for checking Prerequisite Skills.
^ Textbook Option Take the Quick Check below.
Quic Check
Graph each inequality on a number line. (Prerequisite Skill)
1. x> - 3 2. x< - 2
3. x < - 5
5. 7>x
4. x > 1
6. - 4 < x
Solve each equation for y. (Prerequisite Skill)
7. y - 3x= 2 8. y+ Ax= - 5
9. 2 x - y 2 = 7 10. y 2 + 5 = -3x
11. 9 + y 3 = _ x 1 2 . y 3 - 9 = 1U
13. DONUTS A bakery uses the formula 12D = n, where D is the number of dozens of donuts and n is the total number of donuts sold to determine how many dozens of donuts were sold. Solve the equation for D, and determine how many dozens of donuts were sold if 306 donuts were sold. (Prerequisite Skill)
Evaluate each expression given the value of the variable.
14. 3 y - 4 , y = 2 15. 2b+7,b=-3
16. x2 + 2 x - 3 , x = -4a 17. 5z- 2z2 + 1, z = 5/
18. - 4 c 2 + 7 , c = 7 a 2 19. 2 + 3 p 2 , p = - 5 + 2n
20. TEMPERATURE The formula C= | ( F - 32), where C represents a 9
temperature in degrees Celsius and Fin degrees Fahrenheit, can be used to convert between the two measures. If the temperature on a thermometer reads 73°F, what is the temperature in degrees Celsius rounded to the nearest tenth? (Prerequisite Skill)
2 Online Option Take an online self-check Chapter Readiness Quiz at connectED.mcaraw-t .com.
NewVocabulary English Espanol
interval notation p. 5 notacion del intervalo
function p. 5 funcion
function notation p. 7 notacion de la funcion
implied domain p. 7 dominio implicado
zeros p. 15 ceros
roots p. 15 ralz
even function p. 18 funcion uniforme
odd function p. 18 funcion impar
limit p. 24 h'mite
end behavior p. 28 comportamiento final
increasing p. 34 aumento
decreasing p. 34 el disminuir
constant p. 34 constante
maximum p. 36 maximo
minimum p. 36 minimo
extrema p. 36 extrema
secant line p. 38 linea secante
parent function p. 45 funcion del padre
transformation p. 46 transformacion
reflection p. 48 reflexion
dilation p. 49 dilatacion
composition p. 58 composicion
i Vocabulary parabola p. P9 parabola the graph of a quadratic function
slope • Prerequisite Skill • linea pendiente the ratio of the change in y-coordinates to the change in x-coordinates
3
You used set notation to denote elements, subsets, and complements. (Lesson 0-1)
NewVocabulary set-builder notation interval notation function function notation independent variable dependent variable implied domain
function relevant domain
Describe subsets of real numbers.
I Identify and evaluate • functions and state their domains.
Many events that occur in everyday life involve two related quantities. For example, to operate a vending machine, you insert money and make a selection. The machine gives you the selected item and any change due. Once your selection is made, the amount of change you receive depends on the amount of money you put into the machine.
1 Describe Subsets of Real Numbers Real n u m b e r s are u s e d t o descr ibe q u a n t i t i e s s u c h as m o n e y a n d distance. T h e set o f rea l n u m b e r s M i n c l u d e s the f o l l o w i n g subsets o f n u m b e r s .
/Concept Real Numbers
Real Numbers1
Letter Set Examples
Q rationals 0.125, - | , | = 0.666... 0 o
i irrationals V3 = 1.73205...
z integers -5,17,-23,8
w wholes 0,1,2,3...
N naturals 1,2,3,4...
These a n d o t h e r sets of rea l n u m b e r s can be d e s c r i b e d u s i n g s e t - b u i l d e r n o t a t i o n . Se t -bui lder notation uses the p r o p e r t i e s of the n u m b e r s i n the set t o d e f i n e the set.
fx | - 3 < x < 16, x € Z} / J
The set of numbers xsuch that...
x has the given properties...
and x is an element of the given set of numbers.
Use Set Builder Notation
D e s c r i b e the set of n u m b e r s u s i n g set -builder notation.
a. {8,9,10,11,...} T h e set i n c l u d e s a l l w h o l e n u m b e r s greater t h a n or e q u a l t o 8.
[x | x > 8, x G W } Read as the set of all x such that x is greater than or equal to 8
, inmont nf tho cpjnf whole numbers. and x is an element of the set or wnoiu uumuw.
[X | x ^ o,
b. x<7 Unless o t h e r w i s e s tated, y o u s h o u l d assume t h a t a g i v e n set consist^, of rea l n u m b e r s .
There fore , the set i n c l u d e s a l l r ea l n u m b e r s less t h a n 7. {x \ < 7, x G K)
c. a l l m u l t i p l e s of t h r e e
T h e set i n c l u d e s a l l in tegers t h a t are m u l t i p l e s of three, {x \ = 3n , n G Z)
GuideiiPractice
1A. { 1 , 2 , 3 , 4 , 5 , . . . . } 1B. x < - 3 1C. a l l m u l t i p l e s of 7T
4 | Lesson 1-1
flip * You can review ion, including unions
:ions of sets, in
I n t e r v a l n o t a t i o n uses i n e q u a l i t i e s to descr ibe subsets of rea l n u m b e r s . T h e s y m b o l s [ or ] are u s e d t o i n d i c a t e t h a t a n e n d p o i n t is i n c l u d e d i n the i n t e r v a l , w h i l e the s y m b o l s ( or ) are u s e d t o i n d i c a t e
>that a n e n d p o i n t is n o t i n c l u d e d i n the i n t e r v a l . T h e s y m b o l s oo, p o s i t i v e i n f i n i t y , a n d — oo, n e g a t i v e i n f i n i t y , are u s e d t o descr ibe the u n b o u n d e d n e s s of a n i n t e r v a l . A n i n t e r v a l is unbounded i f i t goes o n i n d e f i n i t e l y .
W r i t e each set o f n u m b e r s u s i n g i n t e r v a l n o t a t i o n .
a. - 8 < x < 1 6 ( - 8 , 1 6 ]
b. x < l l ( - 0 0 , 1 1 )
C. x < —16 o r x > 5 (—00, —16] U (5, 00) u read as union
rGuldedPractice 2A. - 4 < y < - 1 2B. a > - 3 2C. x > 9 or x < -2
d e f l t i f y F u n c t i o n s Recal l t h a t a relation is a r u l e t h a t relates t w o q u a n t i t i e s . Such a r u l e p a i r s the e lements i n a set A w i t h e lements i n a set B. The set A o f a l l i n p u t s is the domain o f
the r e l a t i o n , a n d set B conta ins a l l o u t p u t s or the range.
Relat ions are c o m m o n l y represented i n f o u r w a y s .
1. Verbally A sentence describes h o w the i n p u t s a n d o u t p u t s are re la ted .
The output value is 2 more than the input value.
2. Numerically A table of va lues or a set o f o r d e r e d p a i r s relates each i n p u t (x -va lue) w i t h a n o u t p u t v a l u e ( y - v a l u e ) .
{ (0 ,2 ) , ( 1 , 3 ) , ( 2 , 4 ) , (3 ,5)}
3. Graphically Po in ts o n a g r a p h i n the c o o r d i n a t e p l a n e represent the o r d e r e d p a i r s .
4. Algebraically A n e q u a t i o n relates the x- a n d y - c o o r d i n a t e s of each o r d e r e d p a i r .
y = x + 2
A f u n c t i o n is a speciai t y p e of r e l a t i o n .
/Tip in and Range In this text, tation for domain and range i D = and R =, respectively.
KeyConcept Function
Words A function /"from set A to set B is a relation that assigns to each element x in set A exactly one element y in set 6.
Symbols The relation from set A to set 6 is a function.
Set A is the domain. D = {1,2,3,4}
Set B contains the range. R = {6,8,9}
Set A Sets
fl|connectED.mcgraw-hill.com |
StudyTip Tabular Method When a relation fails the vertical line test, an x-value has more than one corresponding y-value, as shown below.
- 2 - 4
3 - 1 3 4 5 6
7 9
A n a l ternate d e f i n i t i o n o f a f u n c t i o n is a set of o r d e r e d p a i r s i n w h i c h n o t w o d i f f e r e n t p a i r s h a v e >the same x - v a l u e . I n t e r p r e t e d g r a p h i c a l l y , t h i s means t h a t n o t w o p o i n t s o n the g r a p h o f a f u n c t i o n i n the c o o r d i n a t e p l a n e can l ie o n the same v e r t i c a l l i n e .
Concept Vertical Line Test Words
A set of points in the coordinate plane is the graph of a function if each possible vertical line intersects the graph in at most one point.
Model
StudyTip Functions with Repeated y-Values While a function cannot have more than one y-value paired with each x-value, a function can have one y-value paired with more than one x-value, as shown in Example 3b.
iiiuu i" Identify Relations that are Functions
D e t e r m i n e w h e t h e r each r e l a t i o n represents y as a f u n c t i o n o f x.
a. T h e i n p u t v a l u e x is a s t u d e n t ' s I D n u m b e r , a n d t h e o u t p u t v a l u e y is t h a t s t u d e n t ' s score o n a p h y s i c s e x a m .
Each v a l u e of x c a n n o t be ass igned t o m o r e t h a n one y - v a l u e . A s t u d e n t c a n n o t receive t w o d i f f e r e n t scores o n a n e x a m . There fore , the sentence describes y as a f u n c t i o n o f x.
x | y
- 8 - 5
- 5 - 4
0 - 3
3 - 2
6 - 3
- 4
8x
Each x - v a l u e is ass igned t o exact ly one y - v a l u e . There fore , the table represents y as a f u n c t i o n of x.
A v e r t i c a l l i n e at x = 4 intersects the g r a p h at m o r e t h a n one p o i n t . There fore , the g r a p h does n o t represent y as a f u n c t i o n of x.
y 2 - 2 x = 5
To d e t e r m i n e w h e t h e r th is e q u a t i o n represents y as a f u n c t i o n o f x, so lve the e q u a t i o n f o r y .
y 2 — 2x = 5 Original equation
y 2 = 5 + 2x Add 2xto each side.
y = + V 5 + 2x Take the square root of each side.
T h i s e q u a t i o n does n o t represent y as a f u n c t i o n of x because there w i l l be t w o c o r r e s p o n d i n g y - v a l u e s , one p o s i t i v e a n d one n e g a t i v e , f o r a n y x - v a l u e greater t h a n —2.5.
w GuidedPractice 3A. T h e i n p u t v a l u e x is the area code, a n d the o u t p u t v a l u e y is a p h o n e n u m b e r i n t h a t
area code.
3B. 3C.
- 6 - 7
2 3
5 8
5 9
9 22
8 - 4 5
8x
3D. 3y + 6x = 18
6 1 Lesson 1-1 I Functions
H707-1783) A Swiss mathematician, Euler was i prolific mathematical writer, polishing over 800 papers in his •retime. He also introduced much
of our modern mathematical "elation, including the use of r''X)for the function
I n f u n c t i o n notation, the s y m b o l / ( x ) is r e a d / o f x a n d i n t e r p r e t e d as the value of the function f at x. Because/(x) c o r r e s p o n d s t o the y - v a l u e o f / f o r a g i v e n x - v a l u e , y o u can w r i t e y =f(x).
E q u a t i o n y = —6x
Rela ted F u n c t i o n f(x) = — 6x
Because i t can represent a n y v a l u e i n the f u n c t i o n ' s d o m a i n , x is c a l l e d the i n d e p e n d e n t v a r i a b l e . A v a l u e i n the range o f / i s represented b y the d e p e n d e n t v a r i a b l e , y .
:ind Function Values I f g (x) = x2 + 8x - 24, f i n d each f u n c t i o n v a l u e .
a. g(6)
To f i n d g (6), replace x w i t h 6 i n g (x) = x 2 + 8x - 24.
g(x) = x2 + 8.v — 24 Original function
g ( 6 ) = ( 6 ) 2 + 8(6) - 2 4
= 36 + 48 - 2 4
= 60
b. gi-4x)
g(x) = -v 2 + 8x - 24
g(-4x) = i-4x)2 + 8(-4x)
= 1 6 x 2 - 32x - 24
24
Substitute 6 for x.
Simplify.
Simplify.
Original function
Substitute —4x for x.
Simplify.
C. £ ( 5 c + 4)
g(x) = x1 + 8.v - 24
g(5c + 4) = (5c + 4) 2 + 8(5c + 4) - 24
= 25c 2 + 40c + 16 + 40c + 3 2 - 2 4
= 25c 2 + 80c + 24
GuidedPractice 2x + 3
I f f(x) = — — - , f i n d each f u n c t i o n v a l u e .
Original function
Substitute 5c + 4 for x.
Expand (5c + 4)2 and 8(5c + 4),
Simplify.
xz - 2x + 1
4A. /(12) 4B. f(6x) 4C. / ( - 3 a +
W h e n y o u are g i v e n a f u n c t i o n w i t h a n u n s p e c i f i e d d o m a i n , the i m p l i e d d o m a i n is the set of a l l rea l n u m b e r s f o r w h i c h the express ion u s e d t o d e f i n e the f u n c t i o n is rea l . I n genera l , y o u m u s t e x c l u d e va lues f r o m the d o m a i n of a f u n c t i o n t h a t w o u l d r e s u l t i n d i v i s i o n b y zero or t a k i n g the e v e n r o o t o f a n e g a t i v e n u m b e r .
StudyTip aming Functions You can use
~er letters to name a function ETC its independent variable. - : ' example, f(x) = yjx— 5 and git) = V f - 5 name the same ~--:tion.
Example 5 Find Domains Algebraically
State t h e d o m a i n o f each f u n c t i o n .
2 + x a. fix) =
xl — 7x
W h e n the d e n o m i n a t o r of 2 + x
x2 - 7x is zero , the express ion is u n d e f i n e d . S o l v i n g x 2 — 7x = 0,
the e x c l u d e d va lues f o r the d o m a i n of t h i s f u n c t i o n are x = 0 a n d x = 7. T h e d o m a i n of t h i s f u n c t i o n is a l l rea l n u m b e r s except x = 0 a n d x = 7, or {x | x =fc 0, x ^ 7, x 6 R} .
b. git) = V t - 5
Because the square r o o t of a n e g a t i v e n u m b e r c a n n o t be rea l , t — 5 > 0. There fore , the d o m a i n o f g(t) is a l l rea l n u m b e r s t s u c h t h a t t > 5 or [5, oo).
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c. hix) = 1
yJx2-9
T h i s f u n c t i o n is d e f i n e d o n l y w h e n x2 — 9 > 0. Therefore , the d o m a i n o f h(x) is (-oo, - 3 ) U (3, oo).
GuidedPractice State t h e d o m a i n o f each f u n c t i o n .
5A. f{x) = 5 x ~ 2 — 5B. h(a) = V a 2 - 4 5C. g(x) = 8x
x2 + 7x + 12 \Jlx + 6
Real-WorldLink Robert Pershing Wadlow of Alton, Illinois, was the tallest man recorded in medical history at 8 feet 11.1 inches. Wadlow weighed almost 440 pounds.
Source: Guinness Book of World Records
StudyTip Relevant Domain A relevant domain is the part of a domain that is relevant to a model. Consider a function in which the output is a function of length. It is unreasonable to have a negative length, so the relevant domain is the set of numbers greater than or equal to 0.
A f u n c t i o n t h a t is d e f i n e d u s i n g t w o or m o r e equat ions f o r d i f f e r e n t i n t e r v a l s of the d o m a i n is c a l l e d a p i e c e w i s e - d e f i n e d f u n c t i o n .
eal-World Example 6 Evaluate a Piecewise-Defined Function
HEIGHT T h e average m a x i m u m h e i g h t o f c h i l d r e n i n i n c h e s as a f u n c t i o n o f t h e i r p a r e n t s ' m a x i m u m h e i g h t s i n i n c h e s can b e m o d e l e d b y t h e f o l l o w i n g p i e c e w i s e f u n c t i o n . F i n d t h e average m a x i m u m h e i g h t s o f c h i l d r e n w h o s e p a r e n t s h a v e t h e g i v e n m a x i m u m h e i g h t s . U s e hix), w h e r e x i s t h e i n d e p e n d e n t v a r i a b l e r e p r e s e n t i n g t h e p a r e n t s ' h e i g h t a n d hix) i s t h e d e p e n d e n t v a r i a b l e r e p r e s e n t i n g t h e c h i l d ' s h e i g h t .
hix) = 1.6* - 41.6 i f 63 < * < 66
3* - 132 i f 66 < x < 68 2x - 66 i f x > 68
a. h(67)
Because 67 is b e t w e e n 66 a n d 68, use hix) = 3x — 132 t o f i n d fo(67).
/z(67) = 3x- 132 Function for 66 < x < 68
= 3(67) - 132 Substitute 67 for x.
= 201 - 132 or 69 Simplify.
A c c o r d i n g to t h i s m o d e l , c h i l d r e n w h o s e parents h a v e a m a x i m u m h e i g h t o f 67 inches w i l l a t t a i n a n average m a x i m u m h e i g h t o f 69 inches .
b. h(72)
Because 72 is greater t h a n 68, use hix) = 2x — 66.
fe(72) = 2.v - 66 Function for x > 68
= 2(72) - 66 Substitute 72 for x.
= 144 - 66 or 78 Simplify.
A c c o r d i n g to t h i s m o d e l , c h i l d r e n w h o s e parents h a v e a m a x i m u m h e i g h t of 72 inches w i l l a t t a i n a n average m a x i m u m h e i g h t of 78 inches .
f GuidedPractice
6. SPEED T h e speed v of a vehic fe i n m i l e s per h o u r can be represented b y the f o l l o w i n g p iecewis e f u n c t i o n w h e n t is the t i m e i n seconds. F i n d the speed of the veh ic le at each i n d i c a t e d t i m e .
v(t)=-
A. o (5 )
4i 60
-6t + 1500
i f 0 < t < 15 i f 1 5 < f < 2 4 0 i f 2 4 0 < f < 2 5 0
B. 0(15) C. 0(245)
8 I Lesson 1-1 I Functions
Exercises £ i c h set o f n u m b e r s i n s e t - b u i l d e r a n d i n t e r v a l
i f p o s s i b l e . (Examples 1 and 2)
2. x < - 1 3
4. { - 4 , - 3 , - 2 , - 1 , . . . }
6. - 3 1 < x < 64
8. x < 0 or x > 100
10. x < 61 or x > 67
12. a l l m u l t i p l e s o f 8
l r > V
. x < - 4
i • < x < 99
< - 1 9 o r x > 2 1
• 0 ,0 .25,0 .50,
< - 4 5 or x > 86
. all m u l t i p l e s o f 5 14. x > 3 2
\e w h e t h e r each r e l a t i o n r e p r e s e n t s y as a f u n c t i o n
The i n p u t v a l u e x is a b a n k a c c o u n t n u m b e r a n d t h e o u t p u t v a l u e y is the a c c o u n t balance .
Tr.e i n p u t v a l u e x is the year a n d t h e o u t p u t v a l u e y is the d a v o f the w e e k .
18.
-50 2.11
--: 2.14
-30 2.16
-20 2.17
-10 2.17
0 2.18
x in 0.01 423
0.04 449
0.04 451
0.07 466
0.08 478
0.09 482
3y + 4x = 11
\y = x
20. x 2 = y + 2
22. 4 y 2 + 1 8 = 96*
24- f = y - 6
26.
28.
o Step-by-Step Solutions begin on page R29.
29. METEOROLOGY T h e f i v e - d a y forecast f o r a c i t y is s h o w n . (Example 3)
1
70"F 49"F
2 3
i f
Hi j Lo Hi
'5*F 53T 70T|j F 57 F °FI56 F
a. Represent the r e l a t i o n b e t w e e n the d a y o f t h e w e e k a n d the e s t i m a t e d h i g h t e m p e r a t u r e as a set o f o r d e r e d p a i r s .
b. Is the e s t i m a t e d h i g h t e m p e r a t u r e a f u n c t i o n of the d a y o f the w e e k ? the l o w t e m p e r a t u r e ? E x p l a i n y o u r r e a s o n i n g .
38. DIGITAL AUDIO PLAYERS T h e sales o f d i g i t a l a u d i o p l a y e r s i n m i l l i o n s o f d o l l a r s f o r a f i v e - y e a r p e r i o d c a n be m o d e l e d u s i n g / ( f ) = 2 4 f 2 - 93f + 78, w h e r e t is the year. T h e a c t u a l sales da ta are s h o w n i n the table . (Example 4)
a. F i n d / ( l ) a n d / ( 5 ) .
b. D o y o u t h i n k t h a t the m o d e l is m o r e accurate f o r the earl ier years or the later years? E x p l a i n y o u r reasoning .
Year Sales($)
1 1 million
2 3 million
3 14 million
4 74 million
5 219 million
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Mate the d o m a i n of each funct ion . (Example 5)
39. fix) = 8x + 12
x2 + 5x + 4 40. g(x) =
x + 1 ;2 - 3x - 40
41. g(a) = \J\+a2
5« 43. /(a) = V4fl - 1
45. f{x) = 1 + - 4 T
42. /z(x) = V 6 - x 2
44. g ( x )
46. g ( x )
16
6 + x + 3
47. PHYSICS T h e p e r i o d T o f a p e n d u l u m is the t i m e f o r one cycle a n d can be ca l cu la ted u s i n g the f o r m u l a
T = 2TC \ / w h e r e £ is the l e n g t h o f the p e n d u l u m a n d V 9.8
9.8 is the g r a v i t a t i o n a l acce lerat ion d u e to g r a v i t y i n meters per second s q u a r e d . Is t h i s f o r m u l a a f u n c t i o n o f £ ? I f so, d e t e r m i n e the d o m a i n . I f n o t , e x p l a i n w h y n o t . (Example 5)
Length £
Period T
Find/(—5 ) and/(12) f o r each p i e c e w i s e f u n c t i o n . (Example 6)
48. f{x) - 4 x + 3 i f x < 3
- x 3 i f 3 < x < 3 x 2 + 1 i f x > 8
{ —5x 2 i f x < —6 x 2 + x + 1 i f - 6 < x < 12
0 .5x 3 - 4 i f x > 12
f 2 x 2 + 6x + 4 i f x < - 4 6 - x 2 i f - 4 < x < 12
14 i f x > 1 2
51. /(*)=<
- 1 5 i f x < - 5 Vx + 6 i f - 5 < x < 10
| + 8 i f x > 1 0
52. INCOME TAX Federa l i n c o m e tax f o r a p e r s o n f i l i n g s ingle i n the U n i t e d States i n a recent year can be m o d e l e d usin^ t h e f o l l o w i n g f u n c t i o n , w h e r e x represents i n c o m e a n d T(x) represents t o t a l tax . (Example 6)
f O.lOx i f 0 < x < 7 2 8 5
T(x) = I 782.5 + 0.15x i f 7285 < x < 31,850
1.4386.25 + 0.25* i f 31,850 < x < 77,100
a. F i n d T(7000), T(10,000), a n d 1(50,000).
b. I f a person ' s a n n u a l i n c o m e w e r e $7285, w h a t w o u l d h i s o r her i n c o m e tax be?
53. PUBLIC TRANSPORTATION T h e n a t i o n w i d e use of p u b l i c t r a n s p o r t a t i o n can be m o d e l e d u s i n g the f o l l o w i n g f u n c t i o n . T h e year 1996 is represented b y t = 0, a n d P(t) represents passenger t r i p s i n m i l l i o n s . (Example 6)
Pit) 0.04^
0.35f + 7.6 0.6t + 11.6
0 < t < 5 5 < t < 10
a. A p p r o x i m a t e l y h o w m a n y passenger t r i p s w e r e there i n 1999? i n 2004?
b. State the d o m a i n of the f u n c t i o n .
Use t h e v e r t i c a l l i n e test t o d e t e r m i n e w h e t h e r each g r a p h represents a f u n c t i o n . W r i t e yes o r no. E x p l a i n y o u r r e a s o n i n g .
54. 55.
56.
J
57.
- 4 8x
58. TRIATHLON I n a t r i a t h l o n , athletes s w i m 2.4 m i l e s , t h e n b i k e 112 m i l e s , a n d f i n a l l y r u n 26.2 m i i e s . Jesse's average rates f o r each l e g of a t r i a t h l o n are s h o w n i n the table .
Leg Rate
swim 4 mph
bike 20 mph
run 6 mph
a. W r i t e a p iecewise f u n c t i o n to descr ibe the dis tance D t h a t Jesse has t r a v e l e d i n t e r m s o f t i m e t. R o u n d t t o the nearest t e n t h , i f necessary.
b. State the d o m a i n o f the f u n c t i o n .
59) ECTIONS Descr ibe the set o f p r e s i d e n t i a l e lec t ion years b e g i n n i n g i n 1792 i n i n t e r v a l n o t a t i o n or i n s e t - b u i l d e r n o t a t i o n . E x p l a i n y o u r r e a s o n i n g .
60. CONCESSIONS T h e n u m b e r of s tudents w o r k i n g the
concession s tands at a f o o t b a l l g a m e can be represented
b y / ( x ) = - j T ^ - , w h e r e x is the n u m b e r of t ickets s o l d .
Descr ibe the r e l e v a n t d o m a i n of the f u n c t i o n .
10 Lesson 1-1 I Functions
ATTENDANCE T h e C h i c a g o C u b s f ranchise has b e e n i n existence since 1874. T h e t o t a l season at tendance f o r its h o m e games can be m o d e l e d byf(x) = 70,050x — 137,400,000, w h e r e x represents the year. Descr ibe the re levant d o m a i n o f the f u n c t i o n .
ACCOUNTING A bus iness ' assets, s u c h as e q u i p m e n t , •ear o u t o r depreciate ov er t i m e . O n e w a y t o calculate
d e p r e c i a t i o n is the s t r a i g h t - l i n e m e t h o d , u s i n g the v a l u e of the e s t i m a t e d l i f e o f the asset. Suppose v (t) = 10,440 -290f describes the v a l u e v(t) o f a c o p y m a c h i n e after : m o n t h s . Descr ibe the r e l e v a n t d o m a i n of the f u n c t i o n .
3d/irt) ,/(a + /z),and f(a + h)-f(a)
h i f h + 0.
(x) = - 5 64. m — Vx
1 x + 4
66. f{x) 2 5 — x
— x) = x2 - 6x + 8 68. fix) = -\x +
5 - x) - —x 70. fix) = x 3 + 9
H = 7x - 3 72. fix) = 5 x 2
(x) = x 3 74. fix) = 11
MAIL T h e U.S. Posta l Service requires t h a t envelopes h a v e a n aspect r a t i o ( l e n g t h d i v i d e d b y h e i g h t ) of 1.3 t o 2.5, i n c l u s i v e . T h e m i n i m u m a l l o w a b l e l e n g t h is 5 inches
1
a n d the m a x i m u m a l l o w a b l e l e n g t h is 11— inches .
height
length-
a. W r i t e the area o f the e n v e l o p e A as a f u n c t i o n of l e n g t h I i f the aspect r a t i o is 1.8. State the d o m a i n of the f u n c t i o n .
B. W r i t e the area o f the e n v e l o p e A as a f u n c t i o n of h e i g h t h i f the aspect r a t i o is 2 .1 . State the d o m a i n of the f u n c t i o n .
c. F i n d the area of a n e n v e l o p e w i t h the m a x i m u m h e i g h t at the m a x i m u m aspect r a t i o .
GEOMETRY C o n s i d e r the c ircle b e l o w w i t h area A a n d : : r cumfe re n ce C.
a. Represent the area o f the c ircle as a f u n c t i o n of i ts c i r cumference .
b. F i n d A(0.5) a n d A ( 4 ) .
C. W h a t d o y o u not i ce a b o u t the area as the c i rcumference increases?
D e t e r m i n e w h e t h e r each e q u a t i o n is a f u n c t i o n o f x. E x p l a i n .
77. x = \y\. x = y3
79. tgt MULTIPLE REPRESENTATIONS I n t h i s p r o b l e m , y o u w i l l i n v e s t i g a t e the range of a f u n c t i o n .
a. GRAPHICAL Use a g r a p h i n g ca lcula tor t o g r a p h / ( x ) = xn f o r w h o l e - n u m b e r va lues of n f r o m 1 t o 6, i n c l u s i v e .
V Jff(x) = x2
[-10,10] scl: 1 by [-10,10] scl: 1
b. TABULAR P red ic t the range of each f u n c t i o n based o n the g r a p h , a n d tabula te each v a l u e o f n a n d the c o r r e s p o n d i n g range .
c. VERBAL M a k e a conjecture a b o u t the range of fix) w h e n n is e v e n .
d. VERBAL M a k e a conjecture a b o u t the range o f / ( x ) w h e n n is o d d .
H.O.T. Problems Use Higher-Order Thinking Skills
80. ERROR ANALYSIS A n a a n d M a s o n are e v a l u a t i n g
- . A n a t h i n k s t h a t the d o m a i n o f the f u n c t i o n /(*) = -x~ — <±
is (—00, —2) o r (1 ,1 ) o r (2, 00). M a s o n t h i n k s t h a t the d o m a i n is {x | x —2, x ± 2, x G M). Is e i ther o f t h e m correct? E x p l a i n .
(8?) RITING IN MATH W r i t e the d o m a i n o f
fix) = — — — i n i n t e r v a l n o t a t i o n a n d i n set-J K ' ix + 3)(x + l)ix - 5) b u i l d e r n o t a t i o n . W h i c h n o t a t i o n d o y o u prefer? E x p l a i n .
82. CHALLENGE G (x) is a f u n c t i o n f o r w h i c h G (1) = 1 , G (2) = 2,
G(3) = 3, a
F i n d G (6).
G(3) = 3, a n d G ( x + 1) = G ( * - 2) G ( * - 1) + 1 for % > 3
G(x)
REASONING D e t e r m i n e w h e t h e r each s t a t e m e n t is true or false g i v e n a f u n c t i o n f r o m set X t o set Y. I f a s t a t e m e n t i s fa lse , r e w r i t e i t t o m a k e a t r u e s t a t e m e n t .
83. E v e r y e l e m e n t i n X m u s t be m a t c h e d w i t h o n l y one e l e m e n t i n Y.
84. E v e r y e l e m e n t i n Y m u s t be m a t c h e d w i t h a n e l e m e n t i n X .
85. T w o or m o r e e lements i n X m a y n o t be m a t c h e d w i t h the same e lement i n Y.
86. T w o or m o r e e lements i n Y m a y n o t be m a t c h e d w i t h t h e same e l e m e n t i n X .
WRITING IN MATH E x p l a i n h o w y o u c a n i d e n t i f y a f u n c t i o n d e s c r i b e d as each o f t h e f o l l o w i n g .
87. a v e r b a l d e s c r i p t i o n of i n p u t s a n d o u t p u t s
88. a set of o r d e r e d p a i r s
89. a table o f va lues
90. a g r a p h
91. a n e q u a t i o n
— connectED.mcgraw-hill.com ] 11
F i n d t h e s t a n d a r d d e v i a t i o n o f each p o p u l a t i o n o f data .
95. BASEBALL H o w m a n y d i f f e r e n t 9 -p layer teams can be m a d e i f there are 3 p l a y e r s w h o can o n l y p l a y catcher, 4 p l a y e r s w h o can o n l y p l a y f i r s t base, 6 p l a y e r s w h o can o n l y p i t c h , a n d 14 p l a y e r s w h o can p l a y i n a n y o f the r e m a i n i n g 6 pos i t ions? (Lesson 0-7)
F i n d t h e v a l u e s f o r x a n d y t h a t m a k e each m a t r i x e q u a t i o n t r u e . (Lesson 0-6)
96. y '4x - 3 ' _x_ L y - 2 J
97. "3yv "27 + 6 x ' .10. . 5y .
98. [9 11] = [3x + 3y 2x + 1]
Use a n y m e t h o d t o s o l v e t h e s y s t e m o f e q u a t i o n s . State w h e t h e r t h e s y s t e m i s consistent, dependent, independent, o r inconsistent. (Lesson 0-5)
99. 2x + 3y = 36 4x + 2y = 48
100. 5x + y = 25 lOx + 2y = 50
101.7x + 8y = 30 7x + 16y = 46
102. BUSINESS A u s e d b o o k store sells 1400 p a p e r b a c k b o o k s p e r w e e k at $2.25 p e r b o o k . T h e o w n e r est imates t h a t he w i l l sel l 100 f e w e r b o o k s f o r each $0.25 increase i n p r i c e . W h a t p r i c e w i l l m a x i m i z e the i n c o m e of the store? (Lesson 0-3)
Use t h e V e n n d i a g r a m to f i n d each o f t h e f o l l o w i n g . (Lesson 0-1)
103. A' 104. A U B
105. B n C 106. AC\B ( 12/ \ ( \ A
u I1 1 I J10 6 J I 4, 5 ,BVJ3 / 7
Skills Review for Standardized Tests
107. SAT/ACT A c i r cu lar cone w i t h a base of r a d i u s 5 has b e e n c u t as s h o w n i n the f i g u r e .
W h a t is the h e i g h t o f the smal le r t o p cone? 104
13
T> 96 B 13 D 94
108. REVIEW W h i c h f u n c t i o n is l inear? _ 2 F f{x)
G gix) = 2.7
H fix) = \[9
J gix) = Vx - 1
12 I Lesson 1-1 I Functions
109. L o u i s is f l y i n g f r o m D e n v e r to Dal las f o r a c o n v e n t i o n . H e can p a r k h i s car i n the D e n v e r a i r p o r t l o n g - t e r m l o t or i n the n e a r b y s h u t t l e p a r k i n g fac i l i ty . T h e l o n g - t e r m l o t costs $1 per h o u r o r a n y f r a c t i o n thereof w i t h a m a x i m u m charge of $6 p e r day. I n the s h u t t l e fac i l i ty , he has to p a y $4 f o r each d a y or p a r t of a day. W h i c h p a r k i n g l o t is less expens ive i f L o u i s r e t u r n s after 2 days a n d 3 hours?
A s h u t t l e f a c i l i t y
B a i r p o r t l o t
C T h e y w i l l b o t h cost the same.
D c a n n o t be d e t e r m i n e d w i t h the i n f o r m a t i o n g i v e n
110. REVIEW G i v e n y = 2.24x + 16.45, w h i c h statement best
describes the effect of m o v i n g the g r a p h d o w n t w o units?
F T h e y - i n t e r c e p t increases.
G T h e x - in te rcept r e m a i n s the same.
H T h e x - in te rcept increases.
J T h e y - i n t e r c e p t r e m a i n s the same.
