CHAPTER 1 Functions, Graphs, and Limits


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1 CHAPTER Functions, Graphs, and Limits Section. The Cartesian Plane and the Distance Formula... Section. Graphs of Equations...8 Section. Lines in the Plane and Slope... MidChapter Quiz Solutions... Section. Functions... Section. Limits...8 Section. Continuit... Review Eercises... Chapter Test Solutions... Practice Test... Copright Houghton Mifflin Compan. All rights reserved.
2 CHAPTER Functions, Graphs, and Limits Section. The Cartesian Plane and the Distance Formula Skills Review ( ) + 7 ( ) ( ) + ( ) ( ) + ( 7 ) ( ) + 7 ( ) ± ±, ( ) + ( ) ( ) ( ) + ( ) + ( ) + ( ) ( ) ± ± + 8, 9. + ( ) (, ) (, ) (, ) (, ) (, ) Copright Houghton Mifflin Compan. All rights reserved.
3 Section. The Cartesian Plane and the Distance Formula. (a) (, ) 9. (a) (, ) (, ) (, ), + (, ) (b) d (c) Midpoint, (, ). (a),, (b) d + ( ), (c) Midpoint,,. (a) (b) d ( ) + ( ) (c) Midpoint,, (., ) (.,.) (,.8) (b) d (. ) + ( (.8) ) (c) Midpoint, (.,.) 7. (a) (, ). (a) a b (, 8) ( ) ( ) c + (, ) (b) a + b + 9 c (b) d ( ) + ( ) (c) Midpoint, (, 8). (a) a b ( 7 ) ( ) c + + (b) a + b c Copright Houghton Mifflin Compan. All rights reserved.
4 Chapter Functions, Graphs, and Limits 7. d ( ) + ( ) d + + d Because d + d d the figure is a right triangle., 9. d ( ) + ( ) d d d d (, ) (, 7) d d (, ) + ( ) ( ) + + ( ) ( ) + + ( ) ( ) + + Because d d d d, the figure is a parallelogram.. d ( ) , ( )( ). d ( ) ±. (a) d +, d > d 8 d 8.7 feet 7. (b) A ( )( 8) square feet Answers will var. Let correspond to 99. The number of subscribers steadil increased from 99 to and then steadil decreased from to. 9. From the graph ou can estimate the values to be: (a) March :,7 (b) December :,9 (c) Ma :, (d) Januar 7:,. (a) 99: $9, (b) 99: $, (c) 997: $, (d) : $8, (, ) d (, ) d d (, ) (, ) d Copright Houghton Mifflin Compan. All rights reserved.
5 Section. The Cartesian Plane and the Distance Formula 7. (a) Revenue midpoint +,9 +,9,, 8,.. (a) Revenue estimate for : $8,. million Profit midpoint ,, 9. Profit estimate for : $9. million (b) Actual revenue: $7, million Actual profit: $ million (c) No, the increase in revenue from to is greater than the increase in revenue from to. No, the profit decreased from to and then increased from to. (d) epenses:,9 8. $, million epenses: 7, $,77 million epenses:,9 79. $9, million (e) Answers will var. (b) The larger the clinic, the more patients a doctor can treat. 7. (a) (, ) is translated to ( +, + ) (,. ) (, ) is translated to ( +, + ) (, ). (, ) is translated to ( +, + ) (,. ) (b) Number of ear infections 7 Medium clinic Large clinic Small clinic Number of doctors 9. Midpoint +, + The point onefourth of the wa between (, ) (, ) is the midpoint of the line segment from + + to,, which is (, ) and ,,. The point threefourths of the wa between (, ) (, ) is the midpoint of the line segment from + +, to,, which is and ,,. Thus, ,,,, and +, +. (a) (b) are the three points that divide the line segment joining,, into four equal parts. ( ) and ( ) + 7 7,, +,, + +,, + + 9,, + +,, + +,, Copright Houghton Mifflin Compan. All rights reserved.
6 8 Chapter Functions, Graphs, and Limits Section. Graphs of Equations Skills Review ( ) ( ) ( ) ( ) ( + ) 9 ( ) ( ) ( ) ( ) ( ) 8 ( + ) ( ) 8 + ( ) ( ) ( ) ( )( ) + + ( + )( + ) ( ) ( ) 9 9. (a) This is not a solution point because. (b) This is a solution point because () ( ). (c) This is a solution point because.. (a) This is a solution point because () +. (b) This is not a solution point because +. (c) This is not a solution point because Copright Houghton Mifflin Compan. All rights reserved.
7 Section. Graphs of Equations 9. The graph of is a straight line with intercept,. So, it matches (e). at 7. The graph of + is a parabola opening up with verte at (, ). So, it matches (c). 9. The graph of has a intercept at (, ) and has intercepts at (, ) and (, ). So, it matches (a).. Let. Then,. Let. Then,. intercept: (, ) intercept: (, ). Let. Then, + ( )( ) +,. Let. Then, + intercepts: (, ), (, ) intercept: (, ). Let. Then, ±. Let. Then, ( ). intercepts: (, ), (, ) intercept: (, ) 7. Let. Then, + ±. Let. Then, ( ) ( ). intercept: Because the equation is undefined when,,. the onl intercept is intercept: (, ) 9. Let. Then, ( ) + ( ). Let. Then, +. intercept: (, ) intercept: (, ). + 7, (, ). (, ) (, ) (, ) Copright Houghton Mifflin Compan. All rights reserved.
8 Chapter Functions, Graphs, and Limits. (, ) (, ) 7. + (, ) 9 (, ) 9. (, ) ,. ± ± ± (, ) (, ) (, ) 7. ( ) ( ) ( ) ( ) Since the point (, ) lies on the circle, the radius must be the distance between (, ) and (, ). r ( + ) + ( ) (, ) (, ) Copright Houghton Mifflin Compan. All rights reserved.
9 ( Section. Graphs of Equations Center Midpoint, (, ) r distance from center to an endpoint + ( ) ( ) ( ) ( ) ( ) ( ) (, ) ( ) ( ) ( ) ( ) +. Solving for in the equation + ields, and solving for in the equation ields. Then setting these two values equal to each other, we have. The corresponding value is, so the point of intersection is (, ).. Solving for in the second equation ields and substituting this into the first equation gives + ( ) ( )( ),. The corresponding values are and, so,. the points of intersection are (, ) and 9.. (, ) + ( ) ( ) ( ) ( ) (, ( ( ) ( ) ( ) ( ) B equating the values for the two equations, we have, ±. The corresponding values are,, and, so the points of intersection are,, (, ), and (, ). 9. B equating the values for the two equations, we have + ( + )( ), ±. The corresponding values are,, and, so the points of intersection are (, ), (, ), and (, ). (, Copright Houghton Mifflin Compan. All rights reserved.
10 Chapter Functions, Graphs, and Limits. (a) C.8 +, R 9. (b) C R.8 +, 9., 7. units (c) P R C 9. (.8 +,), 7.. So, units would ield a profit of $.. R C..8 +,.7,,.7, units,. R C ,,, 9 units,, Equilibrium point (, p ) (, ) 9. (a) Model:.79t 8.t +.9t + 8 t Model Actual , The model is a good fit. Answers will var. (b) When t : $8 million 7. (a) Model:.77t + 7.t + 87 Year 7 Salar (b) Answers will var. (c) When t 9: $77. Yes, this prediction seems reasonable The greater the value of c, the steeper the line Intercepts: (.78, ), (.8, ), (,.87 ) Intercept: (,. ) 8 Intercept: (, ) (,.7) Copright Houghton Mifflin Compan. All rights reserved.
11 Section. Graphs of Equations 8. (a) (b). +.t.t t t t t (c) Yes, the model appears to be a good fit for the data. (d) Sample answer: A quadratic model ma be a good fit for the data. (e) A quadratic model that approimates the data is.8t.8t +.8. (f ) (g) Rational model: When t :. +...% Quadratic model: When t : % The rational model should be used to predict future values, because its values keep decreasing as t increases. The quadratic model should not be used because its values increase for t > 9. Copright Houghton Mifflin Compan. All rights reserved.
12 Chapter Functions, Graphs, and Limits Section. Lines in the Plane and Slope Skills Review , m m, m m ( ) ( ) ( ) ( ) 7 + ( ) ( ) 7 ( ) ( + ) ( + ) + +. The slope is m because the line rises one unit verticall for each unit the line moves to the right.. The slope is m because the line is horizontal.. 7. (, ) (9, ) 8 (, ) (, ) The slope is m. The slope is m. 9 Copright Houghton Mifflin Compan. All rights reserved.
13 Section. Lines in the Plane and Slope The equation of this horizontal line is. So, three additional points are (,, ) (,,and ) (, ).... (, ) (, ) The slope is m. ( ) So, the line is horizontal. ( 8, ) ( 8, ) 8 The slope is undefined because m and 8 ( 8) division b zero is undefined. So, the line is vertical. (, ) The slope is m. 8, (, ) (, ) The slope is m The equation of this line is + ( ) 8. So, three additional points are (,, ) ( 9,, ) and (, ).. The equation of the line is 7 ( ) +. So, three additional points are (, ), (, ), and (, ).. The equation of this vertical line is 8. So, three additional points are ( 8, ), ( 8, ), + + So, the slope is So, the slope is 7. So, the slope is, and ( 8, ). m and the intercept is (, )., m and the intercept is (, )., m and the intercept is (, ).. Because the line is vertical, the slope is undefined. There is no intercept.. So, the slope is m, and the intercept is (, ). 7. (, ), The slope is m 8. Copright Houghton Mifflin Compan. All rights reserved.
14 Chapter Functions, Graphs, and Limits 7. The slope of the line is m Using the pointslope form, we have ( ) +... The slope of the line is m is horizontal, and its equation is +.. So, the line (, ) (, ) (, ) 9. The slope of the line is m. Using the pointslope form, we have +. (, ) (, ) (, ). The slope of the line is m. + Using the pointslope form, we have ,,. The slope of the line is m undefined. So, the line is vertical, and its equation is. (, ) (, ) 8 7. The slope of the line is m + Using the pointslope form, we have (, ) (, 8 ) Copright Houghton Mifflin Compan. All rights reserved.
15 Section. Lines in the Plane and Slope 7 9. Using the slopeintercept form, we have (, ). Because the slope is undefined, the line is vertical and its equation is. (, ). Because the slope is, the line is horizontal and its equation is 7. (, 7) 8. Using the pointslope form, we have + ( ) + +. (, ) 9. The slope of the line joining (, ) and (, ). ( ) The slope of the line joining (, ) and (, ) ( ). is is Because the slopes are different, the points are not collinear.. The slope of the line joining (, 7) and (, ) 7. is The slope of the line joining (, ) and (, ) is. Because the slopes are equal and both lines pass through,, the three points are collinear.. Because the line is vertical, it has an undefined slope, and its equation is.. Because the line is parallel to a horizontal line, it has a slope of m, and its equation is. 7. Given line: + 7, m (a) Parallel: m ( ) (b) Perpendicular: m ( ) Using the slopeintercept form, we have (, ) + 7 9, Copright Houghton Mifflin Compan. All rights reserved.
16 8 Chapter Functions, Graphs, and Limits 9. Given line: + 7, m (a) Parallel: m (b) Perpendicular: m , Given line: is horizontal, m (a) Parallel: m ( ) + (b) Perpendicular: m is undefined (, ) Given line: is vertical, m is undefined (a) Parallel: m is undefined, m,, (b) Perpendicular: (, ) 8. 7 (, ) Copright Houghton Mifflin Compan. All rights reserved.
17 Section. Lines in the Plane and Slope (, ), (, ) F ( C ) F.8C + 9 C + or C ( F 9 ) 87. (a) (, ), (, ) m..( t ).t + The slope m. tells ou that the population is increasing b. thousand per ear. (b) When t :. In, the population was about,,. (c) When t : 8.8 In, the population was about,8,8. (d) :,, :,98, The estimates were close to the eact values. (e) The model could possibl be used to predict the population in 9 if the population continues to grow at the same linear rate. 89. (a) The equipment depreciates $ per ear, so the value is t, where t. (b) (c) When t, the value is $.. (d) The value is $ when t.7 ears. 9. (a) Using the points ( 9, 78) and (,,9 ),, m. 9 So, 78 ( t 9) 7 89 t +. (b) When t : 8 In, the personal income was about $8 billion. (c) When t 7:, In 7, the personal income was about $, billion. (d) : $87 billion 7: $,9 billion The estimates were relativel close to the actual values. 9. (a) C +, (b) R (c) P R C +, 7, (d) When,: P, If the compan sells, units, the profit is $,. (e) R C +, 7, The compan must sell units to break even. 9. (a) W +.7S (b) W +.S (c), The lines intersect at (,, ). If ou sell $,, then both jobs would ield wages of $. (d) No. You will make more mone (if sales are $,) w $ than in the offered at our current job job ( w $ ). Copright Houghton Mifflin Compan. All rights reserved.
18 Chapter Functions, Graphs, and Limits 97., +, 7,.77 So, units.,. C, +, 8,. So, units., 99. C 8,7 +, 8, So, 7 units.,. C, +.,., So,, units., 8,. C 7, + 89, 89, 7.8 So, 7 units., MidChapter Quiz Solutions. (a). (a) (, (, ), (, ) ( ( ( (, 8, ( ( ( (b) d ( ) + ( ( ) ) + 9 (c) Midpoint +,, (b) (c) 9 d Midpoint,, 8 Copright Houghton Mifflin Compan. All rights reserved.
19 MidChapter Quiz Solutions for Chapter. (a). (b) d ( ) + ( ) (c) Midpoint,, a + ( ) ( ) b + c + ( ) ( ) ( ) a + b + c. Use the points (, 79) and (, 8 ) Midpoint, (, 79.) The population in was about,79, people.. +, (, ) (, ) (, ) a (, ) b c (, ) ( ) ( ) Because the point (, ) lies on the circle, the radius must be the distance between (, ) and (, ). ( ) ( ) r ( ) ( ) ( ) ( ) (, ( 7 (, ) 9 (, ) 7 Copright Houghton Mifflin Compan. All rights reserved.
20 Chapter Functions, Graphs, and Limits ( ) ( ) ( ) ( ) (, (, ) (, ( (,,, m. Because the slope is, the line is horizontal and its equation is.. C. +, R 7.9 R C ,., 7.8 The compan must sell 7 units to break even.. (, ), (, ) + m + + (, ) (, ) (, ) (, ) ( ). (, ), (, ) m undefined + Because the slope is undefined, the line is vertical and its equation is. 7. Given line:, m (a) Parallel: m ( ) + 7 (b) Perpendicular: m ( ) Let represent, and represent the sales, in thousands of dollars. (, ), ( 7, 8) 8 m 7 ( ) + When t : When t 9: 7 You can predict the sales to be $,, in and $,7, in An equation for the dail cost C in terms of, the number of miles driven, is C (a) (, 8, ), (,,7),7 8, m 7 8, 7( t ) 7t +, (b) When t : 8, In, our salar will be about $8,. Copright Houghton Mifflin Compan. All rights reserved.
21 Section. Functions Section. Functions Skills Review ( ) + ( + ) ( ) + ( + )( + + 9) ( ) + ( + ) ( ) ± is not a function of since there are two values of for some.. + is a function of since there is onl one value of for each.. is a function of since there is onl one value of for each is a function of since there is onl one value of for each. 9. Domain: (, ) 7 Range: [., ). Domain: (,) (, ) Range: {, } Copright Houghton Mifflin Compan. All rights reserved.
22 Chapter Functions, Graphs, and Limits. Domain: (, ). Range: [, ) 7. ( + ) g g ( + + ) ( + ) , Domain: (, ) (, ) Range: (,) (, ) 7. Domain: (, ) Range: (, ) 9. Domain: (, ) Range: (,]. (a) f ( ) ( ) (b) f( ) ( ). (a). (c) f( ) ( ) (b) g g (c) g g ( + ) f f ( + ) ( + ) ( + ) ( + ) ( + ) + ( + ) [ ] [ + ] + ( ) + + +, 9. f( + ) f( ) +. is not a function of.. is a function of.. (a) ( ) ( + ) ( + )( ), ( + )( ) f + g + (b) (c) f g f g (d) f( g ) f (e) g( f ) g( ) 7. (a) f + g f g + + (b) (c) f g +, (d) f( g ) f( ) ( ) (e) + + g f g f, g (a) f g f f () (b) g f() g g (c) g( f) g (d) f( g) f (e) f g f (f), g f g Copright Houghton Mifflin Compan. All rights reserved.
23 Section. Functions f g f +. ( ) ( ) + g( f ) g( + ). f, f f f f g ( 9 ) 9 ( 9 ) f g f. ( ) ( 9 ) 9 ( 9 ) g f g, 9 f g 9. f f f f f f f 8 f.. 7 f f is onetoone. f is not onetoone because f() f(. ) 7. f + f 9, , f f is not onetoone because f( ) f(. ) f f Copright Houghton Mifflin Compan. All rights reserved.
24 Chapter Functions, Graphs, and Limits 9. (a) + (f ) (b) +. (a) Shifted three units to the left: (b) Shifted si units to the left, three units downward, and reflected: ( ) +. (a) (c) (b) d d d The amounts spent in 997,, and were $7. billion, $ billion, and $88.8 billion, respectivel. (d) +. Total Sales R R + R ( 9 8t.8t ) ( 8.78t) t.8 t, t,,,,,, (e) (a) (b).7 +. p ( p).7..7 p.p ( p).7 p 7 p ( ) units Copright Houghton Mifflin Compan. All rights reserved.
25 Section. Functions C + t () t ( ()) Ct Ct 8t + 7 C is the weekl cost per t hours of production. 7. (a) If, then p 9. If, If >, then p 7. Thus, 9, p 9., <. 7, > (b) P p < then p ,, P ( 9.), <., < 7, >, > 7. (a) Revenue R rn.( n 8) n 9n.n (b) n 7 R (c) The revenue increases and then decreases as n gets larger, so it is not a good formula for the bus compan to use. 7. f g 8 Zeros: 9 9, The function is not onetoone. 77. gt () t + t Domain: Zeros: ± The function is not onetoone. 9 9 Zero: t The function is onetoone. Copright Houghton Mifflin Compan. All rights reserved.
26 8 Chapter Functions, Graphs, and Limits Section. Limits Skills Review. f + (a) f (b) f() c c c + (c) f( h) ( h) ( h). f h + h h +, < +, f (a) (b) f f t + t + + (c). f f( + h) f t + + t + + h ( h) ( h) h + h + h h + + h h h h. f f( + h) f ( + h) h h 8 + h 8 h h h. h Domain: (,) (, ) Range: (,) (, ) g. Domain: [, ] Range: [, ] 7. f Domain: (, ) Range: [, ) 8. f Domain: (,) (, ) Range:, ± (, ) Not a function of (fails the vertical line test) ( 7) Yes, is a function of. Copright Houghton Mifflin Compan. All rights reserved.
27 Section. Limits f() ? ( ) f()...? f()...? f().7..7.? (a) f (b) f. (a) g (b) g. (a) f + g f + g c c c (b) f ( g ) f g (c) c c c c f g. (a) f 7 f c g 9 c c (b) f 8 c f c (c) 7. (a) f + (b) f (c) f 9. (a) f + (b) f (c) f. (a) f + (b) f (c) f. does not eist.. ( ) ( ) Copright Houghton Mifflin Compan. All rights reserved.
28 Chapter Functions, Graphs, and Limits ( ) ( ) ( + )( ) + + ( ) + t t does not eist. t + t + t t + t does not eist. t t ( + )( + ) t ( ) + ( ) t + ( ) So, t. + So, f. f. + does not eist. + ( ) f So, f ( + + ) ( + ) t + t t + t t t t + t t + t t t t + t t t t t ( ) + ( ) ( ) t t t t t t t + t ( t) t Copright Houghton Mifflin Compan. All rights reserved.
29 Section. Limits f().7...,. undefined Because f decreases without bound as tends to from the left, the it is f(), undefined Because f + decreases without bound as tends to from the left, the it is does not eist. 9. (a) If p,, C $,., p (b) If C,, then p ( p) p p p 8 8%. (c) p C The cost function increases without bound as p approaches % (a) (b) When.: A 8. When : A 77.9 (c) Using the zoom and trace features, A $78.8. Because, the length of the compounding period, is approaching +, this it represents the balance with continuous compounding. Copright Houghton Mifflin Compan. All rights reserved.
30 Chapter Functions, Graphs, and Limits 7. (a) f() undefined (b) (c) Domain: (, ) (, ) Range: (, e) ( e, ) +.78 Section. Continuit Skills Review..... ( + )( + ) ( )( + ) ( ) ( )( + ) ( )( ) + ( ) ( ) ( ) ( + )( ) ( + )( ) ( + )( ) + 7 ( + 7) ( )( ) ( )( ) ( ) ( ) ( ) ( ) Copright Houghton Mifflin Compan. All rights reserved.
31 Section. Continuit. Continuous; The function is a polnomial.. Not continuous; The rational function is not defined at ±.. Continuous; The rational function's domain is the entire real line. 7. Not continuous; The rational function is not defined at or. 9. Not continuous; The rational function is not defined at ±.. f is continuous on (,) and (, ) because the domain of f consists of all real numbers ecept. There is a discontinuit at because f ( ) is not defined and f does not eist.. f is continuous on (, ) and +, because the domain of f consists of all real numbers ecept. There is a discontinuit at because f ( ) is not defined and f( ) f.. f + is continuous on (, ) because the domain of f consists of all real numbers. 7. f ( + )( ) (,, ) (, ), and is continuous on, because the domain of f consists of all real numbers ecept ±. There are discontinuities at ± because f and f ( ) are f f do not eist. not defined and and 9. f is continuous on ( ) +, because the domain of f consists of all real numbers.. f + on (,, ) 9 ( ),, and is continuous, because the domain of f consists of all real numbers ecept and. There is a discontinuit at and f are not defined and because f and f does not eist and f f( ).. f + is continuous on all intervals of the form (, ), continuous on, (, ), (, ), c c + where c is an integer. That is, f is,,. f is not continuous at all points c, where c is an integer. There c are discontinuities at, where c is an integer, f does not eist. because f c. +, <, is continuous on (, ) because the domain of f consists of all real numbers, f is defined, f eists, and f f( ). 7. f +,, > is continuous on (,] and (, ). discontinuit at because f 9. f There is a does not eist. + is continuous on (, ) and +, because the domain of f consists of all real numbers ecept. There is a discontinuit at f is not defined, and because f does not eist.. f (,. ) an integer, because is continuous on all intervals c c + There are discontinuities at c, where c is f does not eist. c, >. h f( g ) f( ) h is continuous on its entire domain (, ).. Continuous on [, ] because polnomial. 7. Continuous on [, ) and (, ] because f is a f has a nonremovable discontinuit at. Copright Houghton Mifflin Compan. All rights reserved.
32 Chapter Functions, Graphs, and Limits 9. f ( + )( ) +, 9. f has a removable discontinuit at ; Continuous on (,) and (, ). 7. f 8 + ( + ) +, f has a removable discontinuit at ; Continuous on (,) and (, ).. From the graph, ou can see that f eist, so f is not continuous at. does not ( ) From the graph, ou can see that, where c c is an integer, does not eist. So f is not continuous at all integers c.. f + is continuous on (, ).. f is continuous on all intervals of the form c c +,, where c is an integer.. f +, <, 7. f has a nonremovable discontinuit at ; Continuous on (,) and (, ). f on [, ]. ( + ) + appears to be continuous But it is not continuous at (removable discontinuit). Eamining a function analticall can reveal removable discontinuities that are difficult to find just from analzing its graph. f f a a + + So, 8 a and a. From the graph, ou can see that h and h( ) are not defined, so h is not continuous at and. t A 7., t 9. (a) Amount (in dollars) A Years t The graph has nonremovable discontinuities at t,,,,, (ever months). (b) For t 7, 7(. ) 7 A $,79.7. After 7 ears our balance is $,79.7. Copright Houghton Mifflin Compan. All rights reserved.
33 Review Eercises for Chapter. C , >, not an integer.8 +., >, an integer C is not continuous at,,,.. (a),, 7. P., <, > Nonremovable discontinuities at t,,,, ; s is not continuous at t,,,, or. (b) For t, S $,8,78. The salar during the fifth ear is $, Yes, a linear model is a continuous function. No, actual revenue would probabl not be continuous because revenue is usuall recorded over larger units of time (hourl, dail, or monthl). In these cases, the revenue ma jump between different units of time. P P. + + So, P. Because P( ) is defined, P P P( ), continuous at. eists, and the function is Review Eercises for Chapter. 8 (, ) (, ). Distance ( ) + ( ) Midpoint, ( 7, ) (, ) (., ). Matches (a) 7. Matches (b) 9. Distance ( ) + ( ) Midpoint, ( 8, ) 7. P R C. The tallest bars represent revenues. The middle bars represent costs. The bars on the left of each group represent profits because P R C. 9. The translated vertices are: ( +, + ) (, 7) ( +, + ) (, 8) ( +, + ) ( 8, ) Copright Houghton Mifflin Compan. All rights reserved.
34 Chapter Functions, Graphs, and Limits. Bar graph for data: Mtilus Gammarus 7 Littorina Arbacia 7 Nassarius Ma 9.. Let. Then, ( ) ( ) ( ) ( ) + + ±. Let. Then, ( ) ( ) +. intercepts: (, ), (, ) intercept: (, ) + + r. ( ) ( 7 ) + + r 9 + r. ( ) ( ) 7 r ( ) ( ) Center: (, ) ( ) ( ) r: 8 (, ) Solving for in the second equation ields, and substituting this into the first equation gives + ( ) + + ( )( + ),. The corresponding values are and so the points of intersection are (, ) and (, ). Copright Houghton Mifflin Compan. All rights reserved.
35 Review Eercises for Chapter 7. Solving for in the second equation ields, and substituting this into the first equation gives + + ± ± ±. The corresponding values are + and, so the points of intersection are ( +, + ) and (, ).. (a) C +. R.9 (b) C R , or 8 units. + Slope: m intercept: (, ) 9... Slope: m intercept: (, ) Slope Slope. ( ) ( ) + 7. ( ) (.) 7. Slope: m (horizontal line) intercept: (, ) Copright Houghton Mifflin Compan. All rights reserved.
36 8 Chapter Functions, Graphs, and Limits (a) (b) ; slope ( ) + (c) The line through (, ) and (, ) (d) + Slope of perpendicular is (, 7 ), ( 7, 7 ) 7 7 m 7 (a) 7 ( p ) p + 7 (b) If., p has slope units (c) If.,. + 7 units p. Yes, + is a function of.. No, is not a function of. 7. (a) f () () + 7 (b) (c) f f Domain: (, ) Range: (, ) Domain: [, ) Range: [, ) Domain: (, ) Range: (,] 7. (a) f + g ( + ) + ( ) + f g + (b) + (c) f g ( + )( ) + f( ) + (d) g (e) f( g ) f( ) + ( ) (f ) f ( + ) ( ) g f g + + has an inverse b the horizontal line test. f f + does not have an inverse b the horizontal line test ( ) 7 Copright Houghton Mifflin Compan. All rights reserved.
37 Review Eercises for Chapter 9 8. ( )( ) () t t t t t t ( ) ( ) ( ) ( ) does not eist t + t + t + t + t t + + t + t t ( t ) + t t + ( t ) t + + t + [ t + ] t f() f ( + ) (, ) and (, ) is continuous on the intervals because the domain of f consists of all real numbers ecept. There is a discontinuit at f is not defined. because. The statement is false since. The statement is false since.. The statement is false since f is undefined f is continuous on the intervals +, because the domain of f consists (, ) and of all real numbers ecept. There is a discontinuit at f is not defined. because Copright Houghton Mifflin Compan. All rights reserved.
38 Chapter Functions, Graphs, and Limits. f form ( c, c +, ) + is continuous on all intervals of the where c is an integer. There are discontinuities at all integer values c because f ( ) does not eist. c. f, is continuous on the intervals +, > (,) and (, ). because f There is a discontinuit at. f ( ) does not eist. + f a 8 a So, a 8 and a. 7. (a) 8 The function is continuous for all > ecept,, and. There are discontinuities at these values because C, C, Cdo not eist. (b) C ( ).( ) and The cost of making copies is $. 9. Ct (). t +. t, t >, t not an integer +. t, t >, t an integer The function is continuous for all noninteger values of t >.. (a) 9 (b) t 7 8 D (actual) D (model) t 9 D (actual) D (model) (c) When t, D,7.9. In, the national debt will be about $,7.9 billion. Copright Houghton Mifflin Compan. All rights reserved.
39 Chapter Test Solutions for Chapter Chapter Test Solutions. (a) d ( + ) + ( ) (b) (c) + + Midpoint,, + m. (a) d + ( ) (b) (c) Midpoint,, m. (a) d ( ) + ( ) Equilibrium point (, p ) (., ) The equilibrium point occurs when the demand and suppl are each units.. m When : intercept: (, ) (, ) 7. The line is vertical, so its slope is undefined, and it has no intercept. (b) (c) Midpoint,, m m. When : intercept: (,. ). + ( ) ( ) ( ) ( ) + 9 (,.) 8 8 (, ) Copright Houghton Mifflin Compan. All rights reserved.
40 Chapter Functions, Graphs, and Limits 9. (a) (b) Domain: (, ) Range: (, ) (c). (a). (a) (d) The function is onetoone. (b) Domain: (, ) 9 Range:, ) (c) (d) The function is not onetoone. (b) Domain: (, ) Range: [, ) (c) (., ) f () (, ) (, ) (, ) (, ) f () (, ) (, ) (, ) f () (d) The function is not onetoone f ( ) ( ) f f f ( ) + + f f f + ( ) ( ) f 8 ( ) ( + ) f f f 8 8 ( ) ( ) ( 8 ) f f f ( 8 ) 8 8 ( 8 ) f () , f does not eist. Copright Houghton Mifflin Compan. All rights reserved.
41 Chapter Test Solutions for Chapter. 7. ( )( + ) f () f ().7... f. (a) f + + So, f. Because f is defined, f ( ) f f( ), interval (, ). eists, and the function is continuous on the t (actual) f is continuous on the intervals (,) and (, ) because the domain of f consists of all real numbers ecept. There is a discontinuit at f is not defined. because 9. f is continuous on the interval (, ) because the domain of f consists of all >. (model) 7 t (actual) 7 (model) The model fits the actual data ver well. (b) When t 9: 7. In 9, the number of farms will be about,7,. Copright Houghton Mifflin Compan. All rights reserved.
42 Chapter Functions, Graphs, and Limits Practice Test for Chapter. Find the distance between (, 7) and (, ).. Find the midpoint of the line segment joining (, ) and (, ).. Determine whether the points (, ), (, ), and (, ). Find so that the distance between (, ) and (,). Sketch the graph of.. Sketch the graph of. 7. Sketch the graph of. are collinear. is Write the equation of the circle in standard form and sketch its graph Find the points of intersection of the graphs of + and.. Find the general equation of the line passing through the points ( 7, ) and (, ).. Find the general equation of the line passing through the point (, ) with a slope of m. Find the general equation of the line passing through the point (, 8) with undefined slope.. Find the general equation of the line passing through the point. Given f (a) f () (b) f ( ) (c) f( ) (d) f ( + ), find the following.. Find the domain and range of f.. Given f + and g (a) f ( g ) (b) g( f( )) 7. Given f +, find f. 8. Find ( )., find the following., and perpendicular to the line given b Find. Copright Houghton Mifflin Compan. All rights reserved.
43 Practice Test for Chapter. Find. Find Find f, where f. Find the discontinuities of f +,. +, > 8. Which are removable?. Find the discontinuities of f. Which are removable?. Sketch the graph of f +. Graphing Calculator Required. Solve the equation for and graph the resulting two equations on the same set of coordinate aes Use a graphing calculator to graph f 9 and find f. Is the graph displaed correctl at? Copright Houghton Mifflin Compan. All rights reserved.
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