CHAPTER 2 Functions and Their Graphs

Transcription

1 CHAPTER Functions and Teir Graps Section. Linear Equations in Two Variables Section. Functions Section. Analzing Graps of Functions Section. A Librar of Parent Functions Section. Transformations of Functions Section. Combinations of Functions: Composite Functions... Section.7 Inverse Functions Review Eercises Problem Solving Practice Test

2 CHAPTER Functions and Teir Graps Section. Linear Equations in Two Variables You sould know te following important facts about lines. Te grap of m b is a straigt line. It is called a linear equation in two variables. Te slope (steepness) is m. Te -intercept is 0, b. Te slope of te line troug, and, is m cange in rise cange in run. If m > 0, te line rises from left to rigt. If m 0, te line is orizontal. If m < 0, te line falls from left to rigt. (d) If m is undefined, te line is vertical. Equations of Lines Slope-Intercept Form: m b Point-Slope Form: m Two-Point Form: (d) General Form: A B C 0 (e) Vertical Line: a (f) Horizontal Line: b Given two distinct nonvertical lines L : m b and L : m b L is parallel to L if and onl if m m and b b. L is perpendicular to L if and onl if m m. Vocabular Ceck. linear 7. A B C 0 (iii) general form. slope a (i) vertical line. parallel b (v) orizontal line. perpendicular (d) m b (ii) slope-intercept form. rate or rate of cange (e) m (iv) point-slope form. linear etrapolation 9

3 Section. Linear Equations in Two Variables 9. m. Since te slope is positive, te line rises.. m 0. Te line is orizontal. Matces L. Matces L. m. Because te slope is negative, te line falls. m is undefined. Te line is vertical. Matces L. Matces L. m. Te line falls. Matces L. m. Because te slope is positive, te line rises. Matces L... (, ) m = 0 m = undefined m = m = m = m = (, ) m =. Two points on te line: 0, 0 and,. Te line appears to go troug, 0 and,. Slope rise run Slope 0 7. Two points on te line: 0, and, 0. Te line appears to go troug 0, 7 and 7, 0. Slope rise run Slope Slope: m -intercept: 0, Slope: m -intercept: 0, 0 Slope: m -intercept: 0, (0, ) 0 7 (0, ) 0 (0, 0) 7.. Slope: m -intercept: 0, (0, ) , vertical line Slope: undefined No -intercept Slope: m 0 -intercept: 0, ( ) 0,

4 9 Capter Functions and Teir Graps Slope: m 7 -intercept: 0, (0, ) Slope: m -intercept: 0, (0, ) 0, orizontal line Slope: m 0 -intercept: 0, (0, ) Slope: m 0 -intercept: 0, (0, ) 0 0. Slope: undefined (vertical line) No -intercept 7 0 Slope: undefined (vertical line) -intercept: none. m. Slope (, ). m m is undefined. (, ) 0 (, ) (, ) (, ) (, ). Slope (, 0) 0 (0, 0). m 7 (, ( (, (

5 Section. Linear Equations in Two Variables 97. Slope 7 ( ( 7,, ( 7. m Slope ( (.,.) (.,.) (.7,.) 9 (.,.) 9. Point:,, Slope: m 0 Since m 0, does not cange. Tree points are 0,,,, and,. 0. Because m is undefined, does not cange. Tree oter points are:, 0,,,,.. Point:,, Slope: m. Because m, decreases b for ever one unit increase in. Tree oter points are: 0,, 9,, Since m, increases b for ever one unit increase, 7. in. Tree points are,, 7,, and,.. Point:,, Slope is undefined.. Because m 0, does not cange. Tree oter points are:,,,, 0,. Since m is undefined, does not cange. Tree points are, 0,,, and,.. Point:,, Slope: m. Because m, decreases b for ever one unit Since m, increase in. Tree oter points are:,,,, increases b for ever one unit,. increase in. Tree additional points are,,,, and, Point: 7,, Slope: m. Because m, decreases b for ever unit Since m, increase in. Tree oter points are:,,, 7, increases b unit for ever two unit, 9. increase in. Tree additional points are 9,,, 0, and,. 9. Point 0, ; m 0. Point 0, 0; m 0 (0, ) (0, 0). Point, ; m. Point 0, 0; m (, ) 0 0 (0, 0)

6 9 Capter Functions and Teir Graps. Point, 0; m. Point, ; m 0 (, 0) 0 7 (, ). Point, ; m is undefined.. Point 0, ; m is undefined. Te line is vertical. Because te slope is undefined, te line is a vertical line passing troug 0, wic is te equation. (, ) ( 0, ) 7. Point, ; m 0 Te line is orizontal. ), ). Point, ; m (, ( 9. Point.,.; m 0. Point.,.; m (.,.) (.,.) 0., and,. (, ) (, ),,, (, ) (, ) 7

7 Section. Linear Equations in Two Variables 99., and, 7. Since bot points ave, te slope is undefined, and te line is vertical. (, 7) (, ) 0,,, 0 0 (, ) (, )., and,.,,, (, ( ( ), (, ) ( ), 7. 0, and 9 0, ( 0, ( ( 9 9, 0 (,,, ( 7, ( (, ( , 0. and, 0. 0., 0.,, (, 0.) (, 0.) (, 0.) 0 or 0.. (,.)

8 ( 00 Capter Functions and Teir Graps., and 0 Te line is orizontal.,., (, ) ) ),,, 0 0 (, ) (,. 7, and 7,. m Te line is vertical. 9 and is undefined. 0 ) 7, ) 7 ).,,., Te slope is undefined. Te line is vertical. (., 0.) (., ) 7 7, ).. L : 0,,, 9 Slope of L : L : 0,,, Slope of L : L and m 9 0 m 0 are perpendicular. L. L :,,, m L :,,, m Te lines are neiter parallel nor perpendicular. 7. L :,,, 0 Slope of L : L : 0,,, 7 m 0 7 Slope of L m 0 : L and L are parallel.. L :,,, 9. m L :,,, m Te lines are perpendicular Slope: m,, m,, m 7 Slope: m m,, m,,

9 Section. Linear Equations in Two Variables m 0 Slope: m, 0 and m 0, 7, m 7, 7, m 7 Slope: m m, 7, , 0, m is undefined m, 7, Slope: m 0 m 0,, 0 0 Te reciprocal of 0 is undefined. Te line is vertical, passing troug,. m is undefined.,, m is undefined. Te line is vertical, passing troug,.,, m Slope: undefined Te original line is te vertical line troug. Te line parallel to tis line containing, is te vertical line. A perpendicular to a vertical line is a orizontal line, wose slope is 0. Te orizontal line containing, is te line. Slope: m.,., m....,., m Slope: m.9,., m ,., m

10 0 Capter Functions and Teir Graps , 0, 0, , 0, 0,. 0 c c, c 0. d, 0, 0, d,, c c c 0 d d d d d 0. and are perpendicular is parallel to. is perpendicular to and and are parallel. is perpendicular to and.. is parallel to. is perpendicular to and Set te distance between, and, equal to te distance between, and,. (, ) Tis line is te perpendicular bisector of te line segment connecting, and,. (, ) (, )

11 Section. Linear Equations in Two Variables Set te distance between, and, equal to te distance between, and, (, ) 0 7 (, (, ) ( 9. Set te distance between, and, equal to te distance between 7, and, Tis line is te perpendicular bisector of te line segment connecting, and 7,. (, 7 ( 7, ) 9. Set te distance between, and, equal to te distance between and, , ( (, ( (, ( 7 (, (, ( ( 9. m. Te sales are increasing units per ear. 9. m 00. Te revenues are increasing 00 units per da. m 0. Tere is no cange in sales during te ear. m 0. Te sales are decreasing 0 units per ear. m 00. Te revenues are increasing 00 units per da. m 0. Tere is no cange in revenue during te da. (Revenue remains constant.) 9. 0,,7,,,7: m,,7,,,99:,,99,, 9,77: Te average salar increased te most from 990 to 99 and te least from 99 to 99. CONTINUED m m,7,7 0,99,7 9,77,99 0., 9,77,, 7,0:, 7,0, 0, 79,9: 0, 79,9,,,9: m m m 7,0 9,77 79,9 7,0 0,9 79,

12 0 Capter Functions and Teir Graps 9. CONTINUED,9,7 0,,7,,,9: m $. 0 Te average salar for senior ig scool principals increased b $. per ear over te ears between 990 and Te greatest increase of $. million is between 00 and 00. Te least increase of $. million is between 000 and 00. Slope Eac ear te net profit increases b $9. million feet and m Vertical measurements Horizontal measurements (d) Since m, for ever cange in te orizontal measurement of units, te vertical measurement decreases b. (e) % grade 99., 0, m 00. V 0 t V 0 t V t, 0 t,, m.0 V.0t V.0t. V.t. 0. Matces grap. Te slope is 0, wic represents te decrease in te amount of te loan eac week. Te -intercept is 0, 00, wic represents te original amount of te loan. 0. Matces grap. Te slope is, wic represents te increase in te ourl wage for eac unit produced. Te -intercept is 0,.,wic represents te ourl rate if te emploee produces no units. 0. Matces grap. Te slope is 0., wic represents te increase in travel cost for eac mile driven. Te -intercept is 0, 0, wic represents te fied cost of $0 per da for meals. Tis amount does not depend on te number of miles driven. 0. Matces grap (d). Te slope is 00, wic represents te amount b wic te computer depreciates eac ear. Te -intercept is 0, 70, wic represents te original purcase price.

13 Section. Linear Equations in Two Variables 0 0., 0.,,.0:.0 0. m t 9 represents 999, 9, t t represents 00,, t m For 00, use t : $. N 79.t 0,. For 00, use t 0: 0 $7. t represents 00: N 79. 0,. 9. stores t 0 represents 00: N ,.. stores Tese answers are not reasonable because te are negative. 07. Using te points 0, 7 and, 0, were te first coordinate represents te ear t and te second coordinate represents te value V, we ave 0. 0,,000 and 0, m m 0 7 V 00t,000, 0 t 0 V 7t 7, 0 t , 0,7,,,9: For 00, use t :,007 students. m,9 0, t 0,7 79. For 00, use t 0: 0, students. Te slope is m 79., wic represents te increase in te number of students eac ear. 0. Average annual salar cange from 990 to 00:,7, 0, 9 students per ear m 9, b,, so Nt 9t,. Te slope, 9, represents te average annual cange in enrollment. Using to estimate te enrollment in: 99:, 9 0,7 students 99:, 9,00 students 00:, 9 7,79 students (d) Answers will var.. Sale price List price % of te list price. W S L 0.L S 0.L. C,00.t.0t.7t,00 P R C 7t.7t,00 0.t,00 R 7t (d) 0 0.t,00,00 0.t t ours

14 0 Capter Functions and Teir Graps. 0, 0 and, 7 m m 0 p 0 0 p p units 9 9 units m 0 (d) Since m, eac -meter increase in will increase b meters.. W 0.07S C 0.m 0. Median salar (in tousands of dollars) Using a calculator, te linear regression line is Coosing te points 7, 0 and 0, 00: m 0 Year ( 99) Te answer varies depending on te points cosen to estimate te line. t 9. and Cellular pone subscribers (in millions) Year (0 990) Answers will var. Find two points on our line and ten find te equation of te line troug our points. Sample answer:.7.0 (d) Answers will var. Sample answer: Te -intercept sould represent te number of initial subscribers. In tis case, since b is negative, it cannot be interpreted as suc. Te slope of.7 represents te increase in te number of subscribers per ear (in millions). (e) Te model is a fairl good fit to te data. (f) Answers will var. Sample answer: million subscribers in and Average test score Average quiz score Two approimate points on te line are 0, 9 and 9, 9. m (d) (e) Eac point will sift four units upward, so te best-fitting line will move four units upward. Te slope remains te same, as te new line is parallel to te old, but te -intercept becomes 0, 9 0,, so te new equation is.

15 Section. Linear Equations in Two Variables 07. False. Te slope wit te greatest magnitude corresponds to te steepest line.. Using te Distance Formula, we ave AB, BC 0, and AC. Since 0, te triangle is a rigt triangle.., and, : m 7 0, and 7, 7 : m False, te lines are not parallel On a vertical line, all te points ave te same -value, so wen ou evaluate m, ou would ave a zero in te denominator, and division b zero is undefined.. No. Te slope cannot be determined witout knowing te scale on te -ais. Te slopes will be te same if te scale on te -ais of is and te scale on te -ais of is. Ten te slope of bot is.. Since te steeper line is te one wit a slope of. >, Te slope wit te greatest magnitude corresponds to te steepest line. 7. Te V-intercept measures te initial cost and te slope measures annual depreciation.. No, te slopes of two perpendicular lines ave opposite signs. (Assume tat neiter line is vertical or orizontal.) 9. is a linear equation wit slope m and -intercept 0,. Matces grap (d). 0. Intercepts:, 0, 0, Matces grap.. is a quadratic equation. Its grap is a. parabola wit verte, and -intercept 0,. Matces grap. Intercepts:, 0,, 0, 0, Matces grap or or 7 0 b ± b ac a ± ± ± No real solution Te square root of 9 cannot be negative Answers will var.

16 0 Capter Functions and Teir Graps Section. Functions Given a set or an equation, ou sould be able to determine if it represents a function. Know tat functions can be represented in four was: verball, numericall, grapicall, and algebraicall. Given a function, ou sould be able to do te following. Find te domain and range. Evaluate it at specific values. You sould be able to use function notation. Vocabular Ceck. domain; range; function. verball; numericall; grapicall; algebraicall. independent; dependent. piecewise-defined. implied domain. difference quotient. Yes, te relationsip is a function. Eac domain value is matced wit onl one range value.. No, it is not a function. Te domain value of is matced wit two output values.. No, te relationsip is not a function. Te domain values are eac matced wit tree range values.. Yes, it is a function. Eac domain value is matced wit onl one range value.. Yes, it does represent a function. Eac input value is matced wit onl one output value.. No, te table does not represent a function. Te input values of 0 and are eac matced wit two different output values. 7. No, it does not represent a function. Te input values of 0 and 7 are eac matced wit two output values.. Yes, te table does represent a function. Eac input value is matced wit onl one output value. 9. Eac element of A is matced wit eactl one element of B, so it does represent a function. Te element in A is matced wit two elements, and of B, so it does not represent a function. Eac element of A is matced wit eactl one element of B, so it does represent a function. (d) Te element in A is not matced wit an element of B, so te relation does not represent a function. 0. Te element c in A is matced wit two elements, and of B, so it is not a function. Eac element of A is matced wit eactl one element of B, so it does represent a function. Tis is not a function from A to B (it represents a function from B to A instead). (d) Eac element of A is matced wit eactl one element of B, so it does represent a function.. Eac is a function. For eac ear tere corresponds one and onl one circulation.. Reading from te grap, f 99 is approimatel million.. ±. ± No, is not a function of. Tus, is not a function of.

17 Section. Functions 09.. Yes, is a function of. ± Tus, isnot a function of. 7.. Yes, is a function of. ± Tus, isnot a function of. 9. ± 0. Tus, is not a function of. Yes, isa function of... Yes, is a function of. or Tus, isnot a function of... Tus, tis is not a function of. 7 or 7 0 is a function of.. f. g 7 f f 9 g g f gs 7 s 7 s s 7. Vr r. V 7 V 7 9 Vr r r r t t t f 0. f f f f f f f. q 9. qt t t q q q 9 is undefined. q0 0 0 q 9 Division b zero is undefined. q

18 0 Capter Functions and Teir Graps. f f f if < f if >. f. f f f f, < 0, 0 f f0 0 f. f,, > f f f 0 7., f,, f 7 f f 9 < >., f 0, < > f 9 f 7 f 0 9. f 0. f f f 0 0 f f g g 0 g g g g f g 0. t t 0. f s s s f f f f f t t 0 s 0 f s. f, 0, > f 0 f f 9 f 0 0 f f 0

19 Section. Functions. 9,, < s f f ± f ± ± ,, or f ±. f g. f g. f g or or 0. f g 0 0 0, wic is a contradiction, since represents te principal square root. 0

20 Capter Functions and Teir Graps 7. f. Since f is a polnomial, te domain is all real numbers. f 9. t t Because f is a polnomial, te domain is all real numbers. Domain: All real numbers ecept t 0 0. s 0. g 0. Domain: f t t Because f t is a cube root, te domain is all real numbers t. Te domain is all real numbers.. f. Domain: 0 0 Critical numbers: ± Test intervals:,,,,, Test: Is 0? Domain: f 0 0 Test intervals:,,, 0, 0, Domain: All real numbers, 0 or, 0,. g f s s s Domain: All real numbers ecept 0, 0 0 Te domain is all real numbers 0 and. Domain: s 0 s and s Te domain consists of all real numbers s, suc tat s and s.. f 9. f 70. f 9 Domain: 0 and Te domain is all real numbers suc tat > or,. Te domain is all real numbers suc tat > 0 or 0,. 9 > 0 > 0 Test intervals:,,,,, Te domain is all real numbers < or >. 7. f 7.,,,, 0, 0,,,, f f f f 0 0 f f,,,, 0,,,,,

21 Section. Functions 7. f 7.,,,, 0,,,,, f,,, 0, 0,,,,, 7. B plotting te points, we ave a parabola, so g c. Since, is on te grap, we ave c c. Tus, g. 7. B plotting te data, ou can see tat te represent a line, or f c. Because 0, 0 and, are on te line, te slope is Tus, f Since te function is undefined at 0, we ave r c. Since, is on te grap, we ave c c. Tus, r. 7. B plotting te data, ou can see tat te represent c. Because and, and te corresponding values are and, and. c 79. f f f 0. f f f f, 0 f f f f 0 0 f 0, 0. f f f f, 0. f f f. g f, 0 g g , 0, 9

22 Capter Functions and Teir Graps. f t f t f t t f t t t t t t t t t, t. f. f f f f f f 7. A and P s P s s. A P P A r, C r r C A C C 9. Heigt, Volume, V Volume V Heigt V is a function of. Te volume is maimum wen and V 0 cubic centimeters. V Domain: 0 < < 90. Te maimum profit is $7. Profit Revenue Cost Profit P Order size Yes, P is a function of. (price per unit)number of units costnumber of units , > , > 00

23 Section. Functions 9. A b Since 0,,,, and, 0 all lie on te same line, te slopes between an pair are equal. 0 0 Terefore, A. Te domain of A includes -values suc tat > 0. Using metods of Section. we find tat te domain is >. (0, ) (, ) (, 0) 9. A l w 9. But, so A, 0 < < feet If te cild olds a glove at a eigt of feet, ten te ball will be over te cild s ead since it will be at a eigt of feet. 9. dt.0t 7, 0 t 7 were t represents 99..7t, 0 t 99: t and d.0 7 billion dollars $,000,000,000 99: t and d billion dollars $7,000,000,000 99: t and d.0 7 billion dollars $,000,000,000 99: t and d billion dollars $7,000,000,000 99: t and d.0 7 billion dollars $,000,000,000 99: t and d billion dollars $7,000,000, : t 7 and d billion dollars $7,000,000,000 99: t and d.7. billion dollars $,00,000, : t 9 and d billion dollars $0,00,000, : t 0 and d0.70 billion dollars $,000,000,000 00: t and d.7.7 billion dollars $,700,000,000 00: t and d.7 0. billion dollars $0,00,000,000

24 Capter Functions and Teir Graps 9. pt 0.t 0.7t 7.,.0t., Year Function Value Price 990 p0 7. $7,00 99 p.0 $,0 0 t 7 t 9. V l w were 0. Tus, 0 and V 0 0. Domain: 0 < < 7, p 9. $9, 99 p 0. $0, 99 p.9 $,9 99 p.7 $, p 7.7 $7,7 997 p7 0.0 $0,0 99 p. $, Te dimensions tat will maimize te volume of te package are. From te grap, te maimum volume occurs wen. To find te dimension for, use te equation p9. $, p0. $,00 00 p. $,00 00 p. $, Cost variable costs fied costs C.0 9,000 Revenue price per unit number of units R 7.9 Profit Revenue Cost P ,000 P. 9, Model: Labels: Total cost C Fied cost 000 Variable costs 0.9 Equation: C C C Total cost Fied costs Variable costs R nrate n n 0, n 0 R.00n 0.0n n n 0 0n n, n 0 0 n Rn $7 $700 $7 $70 $7 $700 $7 Te revenue is maimum wen 0 people take te trip. 00. F F,7.0 9, ,70.9,,7.,79,0 Te force, in tons, of te water against te dam increases wit te dept of te water. It appears tat approimatel feet of water would produce,000,000 tons of force.,000, ,000, feet

25 Section. Functions d d d 000 Domain: d 000 (since bot d 0 and d ft f00 f Te number of treatened and endangered fis species increased, on average, b. per ear from 99 to 00. Year Actual Number Number from te Number from te 99 of Fis Species Algebraic Model Calculator Model , 0,, (d) Te algebraic model is an ecellent fit to te actual data. (e) Te calculator model is It also gives a good fit, but not as good as te algebraic model. 0. False. Te range is,. 0. True. Te set represents a function. Eac -value is mapped to eactl one -value. 0. Te domain is te set of inputs of te function, and te range is te set of outputs. 0. Since f is a function of an even root, te radicand cannot be negative. g is an odd root, terefore te radicand can be an real number. Terefore, te domains of f and g are different. 07. Yes. Te amount tat ou pa in sales ta will increase as te price of te item purcased increases. No. Te lengt of time tat ou stud te nigt before an eam does not necessaril determine our score on te eam. 0. No. During te course of a ear, for eample, our salar ma remain constant wile our savings account balance ma var. Tat is, tere ma be two or more outputs (savings account balances) for one input (salar). Yes. Te greater te eigt from wic te ball is dropped, te greater te speed wit wic te ball will strike te ground. 09. t t 0. t t t t t t t t t t

26 Capter Functions and Teir Graps , and,. m Slope m , and,. m Slope 0 m Section. Analzing Graps of Functions You sould be able to determine te domain and range of a function from its grap. You sould be able to use te vertical line test for functions. You sould be able to find te zeros of a function. You sould be able to determine wen a function is constant, increasing, or decreasing. You sould be able to approimate relative minimums and relative maimums from te grap of a function. You sould know tat f is odd if f f. even if f f.

27 Section. Analzing Graps of Functions 9 Vocabular Ceck. ordered pairs. vertical line test. zeros. decreasing. maimum. average rate of cange; 7. odd. even secant. Domain:,, Range: 0,. Domain:,. Domain:, Range: 0, Range: 0,. Domain:,,,. f 0 f. f f Range:, f 0 (d) f f 0 (d) f 0 7. f f 0. f 0 f 9. f0 (d) f f (d) f A vertical line intersects te grap just once, so is a function of. 0.. A vertical line intersects te grap no more tan once, so is a function of. ± is not a function of. Some vertical lines cross te grap twice... A vertical line intersects te grap more tan once, so is not a function of.. A vertical line intersects te grap just once, so is a function of. A vertical line intersects te grap more tan once, so is not a function of or 0 or f f or 0 f ± 9 0 ± f , 0, ±,,

28 0 Capter Functions and Teir Graps. 0. f Zero: Zeros: 0, 7 f f Zero: 0 Zero: Zeros: 0 ±. 0 Zero: f ± ±.. f f is increasing on,.. f. Te grap is decreasing on, and increasing on,. f f is increasing on, 0 and,. f is decreasing on 0,.. f. Te grap is decreasing on, and increasing on,., f,, f is increasing on, 0 and,. f is constant on 0,. 0 0 < >. f,, > Te grap is decreasing on, 0 and increasing on, and 0,.

29 Section. Analzing Graps of Functions 7. f. Te grap is decreasing on and and, increasing on, and 0, f is increasing on,.. f is constant on,. f is decreasing on,., 0 9. f 0. g Constant on, Increasing on, f g 0. g s s 7. Decreasing on, 0; Increasing on 0, s 0 g s 0 Decreasing on, 0; Increasing on 0, f t t. f Increasing on, 0; Decreasing on 0, t 0 f t 0 Increasing on, 0,, ; Decreasing on,, 0, 0 f 0. f Decreasing on, 0 f 0

30 Capter Functions and Teir Graps. f 7. f 9 Increasing on 0, Increasing on, ; Decreasing on, 0 0 f 0.. f f 0 f Decreasing on, 0; Increasing on 0, 9. f 0. 0 f 7 0 Relative minimum:, 9 7 Relative minimum:, or 0.,.. f. f 9. f 0 Relative maimum:., 0. Relative maimum:., 0. Relative minimum:.,.0 Relative maimum:.79,.. f. 7 7 f f 0 on,. Relative maimum: 0.,.0 Relative minimum:.,.0

31 Section. Analzing Graps of Functions. f 7. f 0 0, f f 0 on, and 0,.. f 9. f f 0 f 0 on,. 0 0, 0,, 0. f. f 0 0 0, f f is never greater tan 0. ( f < 0 for all.). f. f is alwas greater tan 0., f f f Te average rate of cange from 0 to is.. Average rate of cange: f f f f f 9 Te average rate of cange from to is.. Average rate of cange: f f 7 7. f f f 0 Te average rate of cange from to is 0.

32 Capter Functions and Teir Graps. Average rate of cange: f f f f f Te average rate of cange from to is. 70. Average rate of cange: f f 0 7. f f f f is even. -ais smmetr Te function is neiter odd nor even. No smmetr g 7. g g g is odd. Origin smmetr f f f Te function is odd. Origin smmetr 7. f t t t 7. ft t t t t ft, ft f is neiter even nor odd. No smmetr gs s 77. gs s s gs Te function is even. -ais smmetr top bottom 7. top bottom 79. top bottom 0. top bottom. L rigt left. L rigt left. L rigt left 0. L rigt left. L , L 000 wen watts.

33 Section. Analzing Graps of Functions Te model is an ecellent fit. Te temperature is increasing from A.M. until noon 0 to. Ten it decreases until A.M. to 0. Ten te temperature increases until A.M. 0 to. (d) Te maimum temperature according to te model is about.9f. According to te data, it is F. Te minimum temperature according to te model is about.9f. According to te data, it is F. (e) Answers ma var. Temperatures will depend upon te weater patterns, wic usuall cange from da to da. 7. For te average salaries of college professors, a scale of $0,000 would be appropriate. For te population of te United States, use a scale of 0,000,000. For te percent of te civilian workforce tat is unemploed, use a scale of %.. m 0 Wen, te resulting figure is a square. m m 0 0 Range: A m s A Domain: 0 B te Ptagorean Teorem, s s meters. 9. r.9t 0.7t 0.t 0, t r7 r Te average rate of cange from 00 to 007 is $0. billion per ear. Te estimated revenue is increasing eac ear at a rapid pace Te average rate of cange from 99 to 00: f 00 f Te number of foreign students increased at a stead rate of.7 tousand students eac ear. Te five-ear period of least average rate of cange was 99 to 997. f 997 f Te five-ear period of greatest increase was 997 to 00. f 00 f Te least rate of cange was about.0 tousand students from 99 to 997. Te greatest rate of cange was about. tousand students from 997 to 00..

34 Capter Functions and Teir Graps 9. s 0, v 0 9. s t 7t. s t t (d) Te average rate of cange of te eigt of te object wit respect to time over te interval t 0 to t is feet per second. (e) (f) 0 0 s s0 0 s0, m Secant line: 00 t 0 t Te average rate of cange from t 0 to t : s s0 0 (d) Te slope of te secant line troug 0, s0 and, s0 is positive. Te average rate of cange of te position of te object from t 0 to t is feet per second. (e) Te equation of te secant line: m, t. (f) Te grap is sown in feet per second v 0 0, s 0 0 s t 0t s t 9t (d) Te average decrease in te eigt of te object over te interval t to t is feet per second. (e) (f) 0 0 s s s 00, m Secant line: t t 0 Te average rate of cange from t to t : s s 0 (d) Te slope of te secant line troug, s and, s is negative. Te average rate of cange of te position of te object from t to t is feet per second. (e) Te equation of te secant line: m Using, s, we ave t t 0. (f) Te grap is sown in. feet per second 7 0 0

35 Section. Analzing Graps of Functions 7 9. v 0 0, s s t 0 s t (d) On te interval t 0 to t, te eigt of te object is decreasing at a rate of feet per second. (e) (f) 0 0 s s0 0 s0 0, m Secant line: t 0 t Te average rate of cange from t to t : s s (d) Te slope of te secant line troug, s and, s is negative. Te average rate of cange of te position of te object from t to t is feet per second. (e) Te equation of te tangent line: m Using, s, we ave t (f) Te grap is sown in. 0 t. feet per second False. Te function f as a domain of all real numbers. 9. False. An odd function is smmetric wit respect to te origin, so its domain must include negative values. 99. Even. Te grap is a reflection in te -ais. Even. Te grap is a reflection in te -ais. Even. Te grap is a vertical translation of f. (d) Neiter. Te grap is a orizontal translation of f. 00. Yes, te grap of in Eercise does represent as a function of. Eac -value corresponds to onl one -value. 0., 0. If f is even, anoter point is,. If f is odd, anoter point is,., 7 If f is even, anoter point is, 7. If f is odd, anoter point is, 7. 0., 9 0., If f is even, anoter point is, 9. If f is even, anoter point is,. If f is odd, anoter point is, 9. If f is odd, anoter point is,.

36 Capter Functions and Teir Graps 0. (d) (e) (f) All te graps pass troug te origin. Te graps of te odd powers of are smmetric wit respect to te origin and te graps of te even powers are smmetric wit respect to te -ais. As te powers increase, te graps become flatter in te interval < <. 0. Te grap of 7 will pass troug te origin and will be smmetric wit te origin. Te grap of will pass troug te origin and will be smmetric wit respect to te -ais or 0 ±0 0 or ± f f f f 7 7. f 0 f 0 0 f 0 0 f 0 0 0

37 Section. A Librar of Parent Functions 9. f 9. f f f does not eist. Te given value is not in te domain of te function. f f f 7 f 9 9. f 9 f 9 f ( 9 f f 9 9, 0. f, f f f f, 0, 0 Section. A Librar of Parent Functions You sould be able to identif and grap te following tpes of functions: (d) Linear functions like f a b Squaring functions like f Cubic functions like f Square root functions like f (e) Reciprocal functions like f (f) Constant functions like f c (g) Absolute value functions like f () Step and piecewise-defined functions like f You sould be able to determine te following about tese common functions: Domain and range -intercept(s) and -intercept Smmetries (d) Were it is increasing, decreasing, or constant (e) If it is odd, even or neiter (f) Relative maimums and relative minimums

38 0 Capter Functions and Teir Graps Vocabular Ceck. f. f. f (g) greatest integer function (i) identit function () reciprocal function. f. f. f c squaring function square root function (e) constant function 7.. f 9. f a b f (f) absolute value function cubic function (d) linear function. f, f 0. f, f, and 0,,,, m 0 0 m 0 f f f 7. f, f 7, and, 7 m 0 7 f 7. f 9, f, 9,, 9 0 m f 9 f 7 0

39 Section. A Librar of Parent Functions. f, f. f 0, f, and, 0,,, m m f f 7 f f, f, m and, f, f,,, m 7 7 f f f f 0. f. f

40 Capter Functions and Teir Graps. f. f. f 0.. g 7. f 0 0. f 9. f 0. g f. f. g f. f k 9. f f. f.9 9 f. (d) f 7

41 Section. A Librar of Parent Functions 0. g. g g g (d) g (d).. 9. f 7. f f f 7 7 (d) f (d). k. k. k...9 k (d) k. g g g g0.. (d) g... g 7 7. g g g g (d) g g 9. g 0. g

42 Capter Functions and Teir Graps. g. g. f, < 0, 0. g,, >. f, < 0, 0. f,, > f,, 0 >.,, < ,,, < < , k,, < >. s. g Domain:, Range: 0, Sawtoot pattern Domain:, Range: 0, Sawtoot pattern

43 Section. A Librar of Parent Functions. Common function: f. Common function:. Common function: f g g. Common function: 7. Common function: f c. Common function: g 7 9. Common function: f 0. Common function:. C t, t > 0 g C Cost (in dollars) 7 0 Time (in minutes) t C. $.. C t.0 0.t is te appropriate. C 0.7.9, > 0 model since te cost does not increase until after C te net minute of conversation as started. Cost (in dollars) C Time (in minutes) C $7.9 t Cost of overnigt deliver (in dollars) 0 0 Weigt (in pounds) C $0.

44 Capter Functions and Teir Graps. Model: Labels: Total cost C Flat rate 9.0 Rate per pound.0, > 0 Equation: C 9.0.0, > 0 Cost of overnigt deliver (in dollars) C Total cost Flat rate Rate per pound. 7 Weigt (in pounds) W, 0 0, W0 0 $0 W0 0 $0 W 0 $70 W0 0 0 $0 W, 0, 0 < 0 > 0 0 < >. For te first two ours te slope is. For te net si ours, te slope is. For te final our, te slope is. Inces of snow 0 0 Hours t f t t, t, t 0, 0 t < t < t 9 To find f t t, use m and,. t t To find f t use m t 0, and., t t 0 Total accumulation. inces 7. Te domain of f.97. is <. One wa to see tis is to notice tat tis is te equation of a line wit negative slope, so te function values are decreasing as increases, wic matces te data for te corresponding part of te table. Te domain of f is ten. f Revenue (in tousands of dollars) Mont ( Januar) f.97).. Tese values represent te income in tousands of dollars for te monts of Ma and November, respectivel. (d) Te model values are ver close to te actual values. Mont, Revenue, Model, f

45 Section. A Librar of Parent Functions 7. Interval Intake Pipe Drainpipe Drainpipe 0, Open Closed Closed, 0 Open Open Closed 0, 0 Closed Closed Closed 0, 0 Closed Closed Open 0, 0 Open Open Open 0, Open Closed Open, 0 Open Open Open 0, 0 Open Open Closed 9. False. A piecewise-defined function is a function tat is defined b two or more equations over a specified domain. Tat domain ma or ma not include - and -intercepts. 70. True. f, is equivalent to te given piecewise function. 7. For te line troug 0, and, : For te line troug, and, 0: f, 0, < m 0 0 m 0 0 Note tat te respective domains can also be 0 < and. 7. f, 7, > > 9 0 > 0 > or < 7. L :, and, 0 7. L :, 7,, m 0 L :, and, 9 m 9 Te lines are neiter parallel nor perpendicular. m L :,,, 7 m Because te slopes are neiter te same nor negative reciprocals, te lines L and L are neiter parallel nor perpendicular.

46 Capter Functions and Teir Graps Section. Transformations of Functions You sould know te basic tpes of transformations. Let f and let c be a positive real number.. f c Vertical sift c units upward. f c Vertical sift c units downward. f c Horizontal sift c units to te rigt. f c Horizontal sift c units to te left. f Reflection in te -ais. f Reflection in te -ais 7. cf, c > Vertical stretc. cf, 0 < c < Vertical srink 9. fc, c > Horizontal srink 0. fc, 0 < c < Horizontal stretc Vocabular Ceck. rigid. f ; f. nonrigid. orizontal srink; orizontal stretc. vertical stretc; vertical srink. iv ii iii (d) i f c. Vertical sifts c : f c : f c : f unit down unit up units up c = c = c = f c c : f c : f c : f Horizontal sifts unit left unit rigt units rigt c = c = c = f c c : f c : f c : f Horizontal sift four units left and a vertical sift unit down unit up units up c = c = c =

47 Section. Transformations of Functions 9. f c c : f c : f c : f c : f Vertical sifts units down unit down unit up units up c = c = c = c = 0 f c Horizontal sifts c : f c : f c : f c : f units left unit left unit rigt units rigt c = c = c = c = f c Horizontal sift units rigt and a vertical sift c : f c : f units down unit down c = c = c : f c : f unit up units up c = c =. f c c : f c 0 : f c : f Vertical sifts units down Common function units up c = c = 0 c = f c c : f c 0 : f Horizontal sifts units rigt Common function c = c = 0 c = c : f units left f c c : f c 0 : f c : f Horizontal sift unit rigt and a vertical sift units down units up c = c = 0 c =

48 0 Capter Functions and Teir Graps. f c, c, < 0 0 f c, c, < 0 0 c = c = c = c = c = c = c = c = c = c = c = c =. f f f Vertical sift units upward (0, ) (, ) (, ) (, ) Horizontal sift units to te rigt (, ) (, ) (, 0) (, ) Vertical stretc (eac -value is multiplied b ) (, 0) (0, ) (, ) (, ) (d) f Reflection in te -ais (e) f Horizontal sift units to te left (f) f Reflection in te -ais (0, ) (, 0) (, ) (, ) (, 0) (, ) (, ) (0, ) (, ) (, ) (, 0) (0, ) f (g) Horizontal stretc (eac -value is multiplied b ) (, 0) (, ) (, ) 7 9 (0, )

49 Section. Transformations of Functions. f f f Reflection in te -ais 0 (, ) (, ) 0 (0, ) (, ) Vertical sift units upward 0 0 (, ) (, ) (, ) (0, ) Vertical stretc (eac -value is multiplied b ) 0 (, ) 0 (, ) (0, ) (, ) (d) f (e) f (f) f Reflection in te -ais and a orizontal sift units to te rigt Vertical sift units downward Reflection in te -ais and a vertical sift unit downward 0 0 (, ) (, ) (0, ) (0, ) 0 (, ) (0, ) (, ) (, ) (, ) 0 (, ) (, ) (0, ) (g) f Horizontal srink (eac -value is divided b ) (, ) (, ) (, ) (0, ) 7. f f f Vertical sift unit downward Horizontal sift unit to te rigt Reflection about te -ais (, ) (0, ) (, ) (, ) (, ) (0, ) (, ) (, ) (, 0) (, ) (, 0) (, ) CONTINUED

50 Capter Functions and Teir Graps 7. CONTINUED (d) f Horizontal sift unit to te left (e) f Reflection about te -ais and a orizontal sift units to te rigt (f) f Vertical srink eac -value is multiplied b (, ) (, ) (0, 0) (, ) (0, ) (, 0) (, ) (, ) (, ) ) ) 0, (, 0) ) ), (g) f Horizontal srink eac -value is multiplied b (, ) (0, ) (, 0 (, ( (. f Horizontal sift units to te rigt 0 (, 0) (, ) (, 0) 0 (, ) (, ) f Reflection in te -ais and a vertical sift units upward 0 (, 7) (, ) 0 (, ) (0, ) (, 7) f Vertical srink eac -value is multiplied b (, 0) (, 0) (, ( (0, ( (, ( (d) f (e) f (f) f 0 Reflection in te -ais and a orizontal sift unit to te left Reflection in te -ais Vertical sift 0 units downward ( 7, ) (, 0) 0 (, ) (, 0) (, ) (0, ) (, 0) (, 0) (, ) (, ) (0, ) (, 0) (, ) 0 (, 0) (, ) 0 CONTINUED

51 Section. Transformations of Functions. CONTINUED (g) f Horizontal stretc (eac -value is multiplied b ) (0, ) ( 9, 0) (9, 0) (, ) (, ) 9. Vertical sift unit downward f Reflection about te -ais, orizontal sift unit to te left, and a vertical sift unit upward f Reflection about te -ais, orizontal sift units to te rigt, and a vertical sift units upward f (d) Horizontal sift units to te rigt and a vertical sift units downward f 0. Te grap of f was reflected in te -ais and sifted upward unit. Te grap of f was sifted to te rigt unit and upward unit. Te grap of f was reflected in te -ais and sifted to te left units and downward unit. (d) Te grap of f was sifted to te rigt 0 units and downward units. 0. Vertical sift units upward. Te grap of f was sifted down units. f Reflection in te -ais and a orizontal sift units to te left f Horizontal sift units to te rigt and a vertical sift units downward f (d) Reflection in te -ais, orizontal sift units to te rigt, and a vertical sift unit downward f Te grap of f was sifted downward 7 units and to te left unit. 7 Te grap of f was reflected in te -ais and sifted to te rigt units and upward units. (d) Te grap of f was reflected about te - and -ais and sifted to te rigt units and downward units.. Common function: f Horizontal sift units to te rigt: (. Common function: Transformation: vertical srink Formula:. Common function: f Reflection in te -ais:. Common function: Transformation: vertical sift Formula:

52 Capter Functions and Teir Graps 7. Common function: f Reflection in te -ais and a vertical sift unit upward:. Common function: Transformation: orizontal sift Formula: 9. g 0. Common function: f Reflection in te -ais and a vertical sift units upward (d) g f g Common function: f Horizontal sift of units to te rigt (d) g f. g 7. Common function: Vertical sift 7 units upward (d) g f f g Common function: Reflection in te -ais; vertical sift of unit downward (d) g f f. g. g 7 Common function: Vertical srink of two-tirds, and a vertical sift units upward 7 f Common function: Vertical stretc of and a orizontal sift 7 units to te rigt of f 0 f 0 (d) g f (d) g f 7

53 Section. Transformations of Functions. g. Common function: f Reflection in te -ais, orizontal sift units to te left, and a vertical sift units upward (d) g f 7 g 0 Common function: f Reflection in te -ais; orizontal sift of 0 units to te left; vertical sift of units upward (d) g f g. Common function: f Horizontal srink b (d) g f g Common function: Horizontal stretc of, f f (d) g f 9. g 0. Common function: f Horizontal sift unit to te rigt and a vertical sift units upward (d) g f g 0 Common function: f Horizontal sift of units to te left; vertical sift of 0 units downward (d) g f 0. g Common function: f Reflection in te -ais; vertical sift units downward (d) g f g. Common function: f Reflection in te -ais; orizontal sift of units to te left; vertical sift of units upward (d) g f 0

54 Capter Functions and Teir Graps. g Common function: Reflection in te -ais, orizontal sift units to te left, and a vertical sift units upward (d) g f f g 9. Common function: f Reflection in te -ais; orizontal sift of units to te rigt; vertical sift of 9 units upward (d) g f g. Common function: f Reflection in te -ais and a vertical sift units up (d) g f g Common function: Horizontal sift of units to te left; vertical stretc (eac -value is multiplied b ) f (d) g f 0 7. g 9. Common function: f Horizontal sift 9 units to te rigt (d) g f 9 g Common function: Horizontal sift of units to te left; vertical sift of units upward (d) g f f g 7 or Common function: g 7 f Reflection in te -ais, orizontal sift 7 units to te rigt, and a vertical sift units downward (d) g f 7

55 Section. Transformations of Functions 7 0. g. Common function: f Reflection in te -ais; orizontal sift of unit to te left; vertical sift of units downward (d) g f 9 g Common function: Horizontal stretc (eac -value is multiplied b ) and a vertical sift units down (d) g f f. g Common function: f Horizontal srink eac -value is multiplied b ; vertical sift of unit upward (d) g f f moved units to te rigt and units down.. f moved units to te left, 7 units upward, and g reflected in te -ais (in tat order) f 7. f moved units to te rigt.. f moved units to te left, units downward, g and reflected in te -ais (in tat order) f or f 7. moved 0 units up and reflected about te -ais. f. f moved unit to te rigt and 7 units downward g 0 0 f 7 9. f moved units to te left and reflected 0. f moved 9 units downward and reflected in bot in bot te - and -aes. te -ais and te -ais g f 9. f. f Reflection in te -ais and a vertical stretc (eac -value is multiplied b ) g Vertical sift units upward and a vertical stretc (eac -value is multiplied b ) g Vertical srink eac -value is multiplied b Reflection in te -ais and a vertical stretc (eac -value is multiplied b )

56 Capter Functions and Teir Graps. f. g Reflection in te -ais and a vertical srink eac -value is multiplied b Vertical stretc (eac -value is multiplied b ) and a vertical sift units downward g f Vertical stretc (eac -value is multiplied b ) Reflection in te -ais and a vertical srink eac -value is multiplied b. Common function: f Vertical stretc (eac -value is multiplied b ) g. Common function: f Vertical stretc (eac -value is multiplied b ) 9. Common function: f Reflection in te -ais; vertical srink eac -value is multiplied b g 7. Common function: Reflection in te -ais; vertical srink eac -value is multiplied b g 0. Common function: Reflection in te -ais; vertical sift of units downward; vertical stretc (eac -value is multiplied b ) f f. Common function: Horizontal stretc (eac -value is multiplied b ). Common function: f Reflection in te -ais, orizontal sift units to te rigt and a vertical sift units upward g. Te grap of as a orizontal sift of units to te left and a vertical sift of units downward. Common function: f. Te grap of as a orizontal sift of units to te rigt and Reflection in te -ais and a a vertical sift of units upward. vertical sift units downward g. g f g f g f Vertical sift units upward Vertical sift unit downward Reflection in te -ais 7 g g 7 g CONTINUED

57 Section. Transformations of Functions 9. CONTINUED (d) g f Reflection in te -ais and a vertical stretc (eac -value is multiplied b ) (e) g f Horizontal srink eac -value is multiplied b (f) g f Horizontal stretc (eac -value is multiplied b ) g g g 0. g f g f g f Vertical sift units downward Vertical sift unit upward Reflection in te -ais 7 9 g 7 7 g 7 g (d) g f Reflection in te -ais and a vertical stretc (eac -value is multiplied b ) (e) g f Horizontal srink eac -value is multiplied b and a vertical sift unit upward (f) g f Horizontal stretc (eac -value is multiplied b ) and a vertical sift units downward g 9 g 0 g 7. F f t t, 0 t A vertical srink b 0.0 and a vertical sift of 0. units upward Amount of fuel (in billions of gallons) F 0 Year (0 90) t f f Te average increase in fuel used b trucks was 0.77 billion gallons per ear between 90 and 00. gt t 0 f t 0 Tis represents a orizontal sift 0 units to te left. (d) g0. billion gallons Yes. Tere are man factors involved ere. Te number of trucks on te road continues to increase but are more fuel efficient. Te availabilit and te cost of overseas and domestic fuel also plas a role in usage.

58 0 Capter Functions and Teir Graps. Te grap is a orizontal sift 0.9 units to te left of te grap of te common function f and a vertical srink b a factor of M f t 0.00t 0.9 B sifting te grap 0 units to te left, ou obtain represents 990. t 0 Amount of mortgage debt (in trillions of dollars) 7 0 Year (0 990) t 9. True, since, te graps of f and f 70. False. Te point, 7 lies on te transformation. are identical. 7. Te profits were onl as large as epected: 7. If ou consider te -ais to be a mirror, eac of te gt f t -values of te grap of fis te mirror image of eac of te -values of te grap of f. Te profits were $0,000 greater tan predicted: gt f t 0,000 Tere was a two-ear dela: gt f t 7. f Horizontal sift units to te left and a vertical sift unit downward 0, 0,, 0,,,,, 0, 7. Answers will var. is probabl simpler to grap b plotting points and is probabl simpler to grap b translating te grap of ,

59 Section. Combinations of Functions: Composite Functions. 9,. 7,,, 0 7. f f f 7 f 9. f 0 f f 0 f f. f Domain: All real numbers ecept Domain:, or,, Critical numbers: ±9 Test intervals:, 9, 9, 9, 9, Test: Is 0? Solution: 9, 9 f. Domain of f : 9 9 f Domain: All real numbers Section. Combinations of Functions: Composite Functions Given two functions, f and g, ou sould be able to form te following functions (if defined):. Sum: f g f g. Difference: f g f g. Product: fg fg. Quotient: fg fg, g 0. Composition of f wit g: f g fg. Composition of g wit f : g f g f

60 Capter Functions and Teir Graps Vocabular Ceck. addition, subtraction, multiplication, division. composition. g. inner; outer f f 0 g 0 0 g 0 f g f g. 0 f 0 g 0 f g 7. Te domain common to bot functions is,, wic is te domain of te sum. 0 f 0. g f g 0.. f, g. f, g f g f g f g f g f g fg f g f g 7 (d) g f f g Domain: all real numbers ecept fg (d) g f Domain:

61 Section. Combinations of Functions: Composite Functions 7. f, g. f, g f g f g f g f g 9 (d) f g f g fg f g g f f g fg 0 (d) g f Domain: < < Domain: all real numbers ecept 9. f, g 0. (d) f g f g f g f g fg f g g f f g Domain: <, f g f g (d) fg f g f, g Domain: 0 Domain: or. f, g. f, g f g f g f g f g f g f g fg f g fg f f (d) g g, 0 (d) g f Domain: 0, For Eercises, f and g.. f g f g. f g f g 7

62 Capter Functions and Teir Graps. f g0 f 0 g f g f g 7. f gt f t gt t t. 9t t f gt f t gt t t t t t t t 9. fg fg 7 0. fg f g g f f g. g f f 0 0 g g f f g g. g fg f fg f 7. f, g, f g. f + g f g f, g f g f + g f g 7. f, g, f g. f, g f g f f + g g g f f + g

63 Section. Combinations of Functions: Composite Functions 9. f, g, f g f, g, f g 0 f + g f 0 f + g f g 0 g For 0, f contributes most to te magnitude. For >, g contributes most to te magnitude. g contributes most to te magnitude of te sum for 0. f contributes most to te magnitude of te sum for >.. f, g. f g fg f g f g f g f f f f f f, g f g fg f 0 g f g f g f f f f f 9 0. f, g. f g f g f g f g f g f f f f f f, g ( f g f g f g f g f g f f f f f 9. f Domain:. f Domain: all real numbers g Domain: all real numbers g Domain: all real numbers f g f g f f g f g Domain: all real numbers f g f g f Domain of f, g, f g, g f : all real numbers Domain: g g f g f g Domain: all real numbers

64 Capter Functions and Teir Graps 7. f Domain: all real numbers. f Domain: all real numbers g Domain: 0 g Domain: all real numbers f g fg f Domain: 0 g f g f g f g f g f Domain: all real numbers g f g f g Domain: all real numbers Domain: all real numbers f 0. f Domain: all real numbers 9. Domain: all real numbers g Domain: all real numbers g Domain: all real numbers f g fg f Domain: all real numbers g f g f g f g f g Domain: all real numbers Domain: all real numbers g f g f f g Domain: all real numbers. f Domain: all real numbers ecept 0 g Domain: all real numbers f g fg f Domain: all real numbers ecept g f g f g Domain: all real numbers ecept 0. f g Domain: all real numbers ecept ± Domain: all real numbers f g f g f Domain: all real numbers ecept 0 and g f g f g Domain: all real numbers ecept ±. f g f g f f g g 0 0. f g f g fg f g f g fg f 0 g f gf g0. f g f g f g f g f g

65 Section. Combinations of Functions: Composite Functions 7 7. Let f and g, ten f g.. Tis is not a unique solution. One possibilit: Let g and f. f g f 9. Let f and g, ten f g. 0. Tis answer is not unique. 9 One possibilit: Let g 9 and f. f g f 9 9. Let f and g, ten f g.. Tis is not a unique solution. One possibilit: Let g and f. f g f. Let f and g, ten f g. Tis answer is not unique One possibilit: Let g and f g f f T R B. Distance traveled (in feet) T B R Speed (in miles per our) Total sales R R R R 0 t 0.t 0.7t 7 7.t 0.t R 7. ct pt bt dt pt 00 c represents te percent cange in te population in te ear 00.. pt dt ct p represents te number of dogs and cats in 00. t pt dt ct nt nt t represents te number of dogs and cats at time t compared to te population at time t or te number of dogs and cats per capita. 9. At.t 9.t 7, Nt.9t.t 0 A Nt At Nt.t 0.0t Tis represents te combined Arm and Nav personnel (in tousands) from 990 to 00, were t 0 corresponds to 990. A N 0.9 tousand A N. tousand A N 7. tousand A Nt At Nt.t 7.t Tis represents te number of Arm personnel (in tousands) more tan te number of Nav personnel from 990 to 00, were t 0 corresponds to 990. A N. tousand A N. tousand A N. tousand