CHAPTER 1 Functions and Their Graphs

Transcription

1 PART I CHAPTER Functions and Their Graphs Section. Lines in the Plane Section. Functions Section. Graphs of Functions Section. Shifting, Reflecting, and Stretching Graphs Section. Combinations of Functions Section. Inverse Functions Section.7 Linear Models and Scatter Plots Review Eercises Practice Test Houghton Mifflin Compan. All rights reserved.

2 CHAPTER Functions and Their Graphs Section. Lines in the Plane You should know the following important facts about lines. The graph of m b is a straight line. It is called a linear equation. The slope of the line through, and, is m. If m > the line rises from left to right. If m, the line is horizontal. If m <,, the line falls from left to right. (d) If m is undefined, the line is vertical. Equations of Lines Slope-Intercept: m b Point-Slope: m Two-Point: (e) Vertical: a Given two distinct nonvertical lines (d) General: A B c (f) Horizontal: b 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 Check. iii i v (d) ii (e) iv. slope. parallel. perpendicular. linear etrapolation. m. Since the slope is positive, the line rises.. m. The line is horizontal. Matches L. Matches L. m. Because the slope is negative, the line m is undefined. The line is vertical. Matches L. falls. Matches L.. m. The line falls. Matches L. m = m = (, ) m = m =. m. Because the slope is positive, the line rises. Matches L. m is undefined. m = m = (, ) m = Houghton Mifflin Compan. All rights reserved.

3 Section. Lines in the Plane. Slope rise run. The line appears to go through, and,. Slope 7. Slope. Slope (, ) (, ) (, ) (, ) 9. (, ). Slope 7 (, ) (, ) Slope is undefined. (, ). Since m, does not change. Three points are. Since m, does not change. Three additional,,,, and,. points:,,,,,.. Since m is undefined, does not change and the line is vertical. Three points are,,,, and,.. Because m is undefined, does not change. Three other points are:,,,,,.. Since m, decreases for ever unit. Since m, increases for ever unit increase increase in. Three points are,,,, in. Three points are:,,,, and and,.,. Houghton Mifflin Compan. All rights reserved. 7. Since m, increases for ever increase of. Since m, decreases for ever increase of in. Three points are 9,,,, and,. units in. Three points are, 7,,,, Slope: m -intercept:, (, ) 9 Slope: m -intercept:, 9 (, )

4 ) Chapter Functions and Their Graphs Slope: undefined No -intercept Slope: undefined -intercept: none Slope: m -intercept:, ( ),.. Slope: m -intercept:, (, ), ). m,, 7.. m,, (, ) (, ) (, ) Houghton Mifflin Compan. All rights reserved.

5 Section. Lines in the Plane 9.. m undefined. Line is vertical.. horizontal line (, ) (, (. m. Line is horizontal....,,, 7 (.,.) 7. Since both points have, the slope is undefined..,,, 7. Houghton Mifflin Compan. All rights reserved.

6 Chapter Functions and Their Graphs.,,, ,,, ,.,, The slope is 7.. The slope is. Houghton Mifflin Compan. All rights reserved.

7 Section. Lines in the Plane 7. Using the points,, and,,9,. Using the points,, and, 7,, ou have ou have m S, t When t,,9, S t,,. S,, $7,. m S, t When t, 7,, S t,,. S,, $,. 7.. Slope: -intercept:, The graph passes through, and rises unit for each horizontal increase of. Slope: -intercept:, The line slopes downward and passes through the point,. 9.. slope is undefined no -intercept The line is vertical and passes through,. Slope: -intercept:, The line is horizontal and passes through,.... Houghton Mifflin Compan. All rights reserved.. The second setting shows the - and -intercepts more clearl. m L 9 m L m L L and L are perpendicular. (, ) (, ) (, ) (, 9) The first setting shows the - and -intercepts more clearl.

8 Chapter Functions and Their Graphs. L :,,,. (, ) (, ) m L :,,, m (, ) (, ) m L m L L and 7 m L L are parallel. (, ) (, ) (, 7 (, ) ( The lines are neither parallel nor perpendicular.. L :,,, 7. m L :,,, m The lines are perpendicular. Slope: m (, ) (, ) (, ) (, ) Slope: 7 m m,, m,, 7. Slope: m vertical line. slope not defined passes through, Slope: m passes through, and is horizontal m,, 9 Slope: m undefined (vertical line) 9 9 m m,.9,...9. m,.9,...9 Houghton Mifflin Compan. All rights reserved.

9 Section. Lines in the Plane 9. The slope is and, lies on the line. Hence,. The slope is and, lies on the line. Hence,... The slope of the given line is. Then l has slope. Hence,.. The slope of the given line is. Then l has slope. Hence,. 7.. L : ; L : ; L : = = = = = = and are perpendicular. L is parallel to L. and L. L is perpendicular to L = + = L : ; L : ; L : = + = + = = L is parallel to L. is perpendicular to and L. L L Houghton Mifflin Compan. All rights reserved. and are parallel. is perpendicular to and. 7. Years Slope Greatest increase: Greatest decrease: 999.7,.9,,.: Between 99 and, the earnings per share decreased at the rate of.7 per ear. (d) For, and.7.., which is reasonable.

10 Chapter Functions and Their Graphs 7. Years Slope Greatest increase: Smallest increase: ,.,, Between 99 and, the sales (in billions of dollars) increased at the rate of. per ear. (d) For, and...9 (billion), which seems reasonable. 7. rise run 7. The maimum height in the attic is feet. Slope rise run, ft. miles 7.,, m 7. V t V t 79,, m. V.t V.t 9 77.,,, m 7. V, t V t,,,, m V, t V t 7, 79. The slope is m. This represents the decrease in the amount of the loan each week. Matches graph.. The -intercept is. and the slope is., which represents the increase in hourl wage per unit produced. Matches graph.. The slope is m.. This represents the increase. The -intercept is and the slope is, which in travel cost for each mile driven. represents the decrease in the value of the word Matches graph. processor each ear. Matches graph (d). Houghton Mifflin Compan. All rights reserved.

11 Section. Lines in the Plane.,,,, t : V,, V, V, t, t t : etc. V,,7 V t,, t 7 9 V,,7,,,,, 9. Using the points, and,, we have m F 9 C 9 F 9 C. F 9 C Houghton Mifflin Compan. All rights reserved.. F : C : 9 C 7. C F 9 F F F 9: 9 9 C C F 7. 9 C. C 9 C C,.t.t.7t, P R C (d) C : F F : 9 C C 77: R 7t F 9 F 9 C C F 9 77 F. F..t, P 7t.7t,,.t P.t, t hours

12 Chapter Functions and Their Graphs., 7,9,,, students per ear 99 7 p 9: 7,9 7 7,9 students 997: 7,9 77,9 students p : 7,9 9 7, students (Answers could var.) p Let t represent 99. If p,, units. Algebraicall,. If p 9, 9 units. Algebraicall, 9 9., 7,9,,, 7,9, 7,9 t 77 t 7,9 t 7, The slope represents the annual increase in students. It is positive, indicating that Penn State Universit increased its students from 99 to.. Answers will var. The slope is which is equivalent to the rate of change. 9. False. The slopes are different: False. The equation of the line joining, and, 9 is 9. For, a and b are the - and -intercepts. 9 Houghton Mifflin Compan. All rights reserved..

13 9. a 9. b intercepts: Section. Lines in the Plane,,, a and b are the - and -intercepts Intercepts:,,, a b 9 Houghton Mifflin Compan. All rights reserved The slope is negative and the -intercept is negative. Matches.. No. The line does not have an -intercept. a b. Both lines have positive slope, but their -intercepts differ in sign. Matches.. No. cannot be written in slope-intercept form because the slope is undefined. 99. The slope is positive and the -intercept is positive. Matches.. The lines intersect in the first quadrant at a point, where <. Matches.. Yes. Answers will var.

14 Chapter Functions and Their Graphs. Yes. Answers will var. 7. Yes.. Yes. 9. No. The term causes the epression to not be a polnomial.. Yes.. No. This epression is not defined for ±.. No Answers will var. Section. Functions Given a set or an equation, ou should be able to determine if it represents a function. Given a function, ou should be able to do the following. Find the domain. Evaluate it at specific values. Vocabular Check. domain, range, function. independent, dependent. piecewise-defined. implied domain. difference quotient. Yes, it does represent a function. Each domain value is matched with onl one range value.. No, it does not represent a function. The domain values are each matched with three range values.. Yes, the relation represents as a function of. Each domain value is matched with onl one range value. 7. No, it does not represent a function. The input values of and 7 are each matched with two output values.. No, it is not a function. The domain value of is matched with two output values.. Yes, it does represent a function. Ever domain value is matched with onl one range value. 9. Each element of A is matched with eactl one element of B, so it does represent a function.. No, the table does not represent a function. The input values of and are each matched with two different output values.. Yes, the table does represent a function. Each input value is matched with onl one output value. Houghton Mifflin Compan. All rights reserved. The element in A is matched with two elements, and of B, so it does not represent a function. Each element of A is matched with eactl one element of B, so it does represent a function. (d) The element of A is not matched to an element of B, so it does not represent a function.

15 Section. Functions. The element c in A is matched with two elements, and of B, so it is not a function. Each element of A is matched with eactl one element of B, so it does represent a function. This is not a function from A to B (it represents a function from B to A instead). (d) Each element in A is matched with eactl one element of B, so it does represent a function.. Each are functions. For each ear there corresponds one and onl one circulation.. f 7.7 million newspapers. ± Thus, is not a function of. For instance, the values and both correspond to.. ± This is not a function of. For eample, the values and both correspond to... This is a function of. This is a function of Thus, isa function of. This is a function of. 9. ±. ± Thus, is not a function of. For instance, the values and both correspond to. Thus, is not a function of.. This is a function of.. 7does not represent as a function of. All values of correspond to 7.. or Thus, is not a function of.. is a function of, a constant function. Houghton Mifflin Compan. All rights reserved... f f ft t t g g g gt t t t f (d) f c 7. c c ft t f 7 f ft t t 7 (d) g c c c c c c

16 Chapter Functions and Their Graphs. g 7 9. ht t t g 7 7 g h h....7 gs 7 s h 7 s s. Vr r. f V V 7 9 f f... f Vr r r. f f f. q 9 q 9 9 f q 9 is undefined. q 9.. qt t t q q q f f f f t t Division b zero is undefined.. f f f ft t t f is undefined. if t > if t < 7. f, <, f f f Houghton Mifflin Compan. All rights reserved.

17 Section. Functions 7. f,, > 9. f,, > f f f f f f. f,, > f f f., f,, < < f f f 7., f,, f 9 f < < f. ht t t ht Houghton Mifflin Compan. All rights reserved.. f s s s f f f f f. f,, > 9 f s f s. h 9,, h 9 h 9 h h h < h

18 Chapter Functions and Their Graphs 7. f. f 9. f. f 7. f g. f g 7 or or. f. g Since f is a polnomial, the domain is all real numbers. Because g is a polnomial, the domain is all real numbers.. ht t. s Domain: All real numbers ecept t The domain is all real numbers f Domain: all real numbers g Domain: All real numbers ecept, g >.. f. Domain: or h The domain is all real numbers and.. f for numerator, and. for denominator. Domain: >. Houghton Mifflin Compan. All rights reserved. > Domain: all >.

19 Section. Functions 9. f. f Domain:, Range:, g. Domain: all real numbers Range: g 9 Domain:, Range:, Domain: all real numbers Range: 7. f.,,,,,,,,, f,,,,,,,,, 9. f,,,,,,,,, 7. f,,,,,,,,, 7. A r, C r r C Houghton Mifflin Compan. All rights reserved. A C C 7. A bh, in an equilateral triangle b s and: s h s h s s h s s s A s s s h b =s s s

20 Chapter Functions and Their Graphs 7. According to the table, the maimum profit is 7 for. Yes, P is a function of. Profit Revenue Cost price per unitnumber of units costnumber of units 9..., > P,., > 7. From the table, the maimum volume seems to be, corresponding to. Yes, V is a function of. 7 (d) V length width height Domain: < < 7 The function is a good fit. Answers will var A baseheight. Since,,, and, all lie on the same line, the slopes between an pair of points are equal. Therefore, A. The domain is >, since A >. A l w But, so A, < <. Houghton Mifflin Compan. All rights reserved.

21 Section. Functions 77. V lengthwidthheight 7. Cost variable costs fied costs But,, or. C. 9, Thus, V. Revenue price per unit number of units Since > R 7.9 < Profit Revenue Cost < 7. P , Domain: < < 7 P. 9,, 7 The highest point on the graph occurs at. The dimensions that maimize the volume are inches. 79. The domain of.97. is 7. The domain of..7. is. You can tell b comparing the models to the given data. The models fit the data well on the domains above.. f..7..7, which means $, 7 in monthl revenue.. f.97.. $, in monthl revenue for November.. The values obtained from the model are a close fit for the actual data.. nt.t 7.t 77,.9t 7, t corresponds to 99. t < t Houghton Mifflin Compan. All rights reserved.. t 7 9 Model R ratenumber of people.n n.nn 7 n n, n n 9 Rn The revenue increases, and then decreases. The maimum revenue occurs when n. 9 7 The maimum occurs at n.

22 Chapter Functions and Their Graphs. F 9.7 F (Answers will var.) F increases ver rapidl as increases.,, From the table, ft (slightl above ). You could obtain a better approimation b completing the table for values of between and. (d) B graphing F together with the horizontal line,,, ou obtain.7 feet.. f. billion dollars f f billion dollars/ear This is the average earl change from 99 and. t 7 9 Pt (d) The model approimates the data well. P 7. f. g 9. f c f c f h f h c c c c h h h, c f, f h h h h h h h h, h g h h h g h g h h g h g h h h, h Houghton Mifflin Compan. All rights reserved.

23 Section. Functions 9. f f h h h h h h h f h f h h h h f h f h h h h h h h h h h h h h h, h 9. f t, f t 9. f t f t f f 7 7 f t t f7 7 t tt 7, t t 7 7 7, 7 9. False. The range of f is,. 9. True. The first number in each ordered pair corresponds to eactl one second number. 9. f,, > 9. f,, > Houghton Mifflin Compan. All rights reserved., 97. f,, 99. The domain is the set of inputs of the function and the range is the set of corresponding outputs.. An advantage of function notation is that it gives a name to the relationship so it can easil be referenced. When evaluating a function, ou see both the input and output values... < < 9. f,,, < <

24 Chapter Functions and Their Graphs.,, , Section. Graphs of Functions You should be able to determine the domain and range of a function from its graph. You should be able to use the vertical line test for functions. You should be able to determine when a function is constant, increasing, or decreasing. You should be able to find relative maimum and minimum values of a function. You should know that f is Odd if f f. Even if f f. Vocabular Check. ordered pairs. Vertical Line Test. decreasing. minimum. greatest integer. even. Domain: All real numbers. Domain: all real numbers,, Range:, ] Range: all real numbers,, f f. Domain:,. Domain: all real numbers,, Range:, Range:, f f. f. f 7. f 7 Houghton Mifflin Compan. All rights reserved. Domain: All real numbers Range:, Domain:, Range:, Domain: or, Range:,

25 Section. Graphs of Functions. ht t 9.. f f t t 7 Domain:, Range:, 9 Domain: All real numbers Domain:, Range:, Range:,. f Domain: all real numbers f, These are the -intercepts of f. (d) f (e) This is the -intercept of f. (f) (g) (h) f. The coordinates are, f. The coordinates are,. f., f,. Houghton Mifflin Compan. All rights reserved... f Domain: all real numbers f These are the -intercepts of f. (d) f (e) This is the -intercept (and -intercept) of f. (f) f. The coordinates are,. (g) (h),, f. The coordinates are,. f 7., f,. f Domain: all -intercepts (d) (e) -intercept (f) (g) f f f (h) f,,,,,,,

26 Chapter Functions and Their Graphs. f,, > Domain: all f if or -intercepts (d) f (e) -intercept (f) (g) f,, f,, (h) f,,.. A vertical line intersects the graph just once, so is a function of. Graph. ± is not a function of. The vertical line intersects the graph twice. Graph and. 7.. A vertical line intersects the graph more than once, so is not a function of. Graph the circle as. A vertical line intersects the graph just once, so is a function of. Solve for and graph. 9. f. f The graph is decreasing on, and increasing f is increasing on,. on,.. f. f f is increasing on, and,. f is decreasing on,. The graph is decreasing on, and increasing on,.. f. f. f is constant on,. f. The graph is increasing on,. f Houghton Mifflin Compan. All rights reserved. Increasing on, The graph is decreasing on,. Decreasing on,

27 Section. Graphs of Functions 7 7. f. f f. Increasing on, Decreasing on, f is decreasing on,. f Increasing on,, constant on,, decreasing on, 9 The graph is increasing on,, constant on,, and decreasing on,.. f. f. Relative minimum:, 9 Relative minimum:, Relative maimum:, Relative minimum:.,... h Relative minimum:.,. Houghton Mifflin Compan. All rights reserved.. Relative minimum:, 7 Relative maimum:, g, is not a relative maimum because it occurs at the endpoint of the domain,. Maimum:.7,.

28 Chapter Functions and Their Graphs 7. f. f f() = (, 9) (, ) f() = Minimum:, 9 Relative minimum:, Relative minimum:, Answers are the same. Minimum:, 9 Answers are the same. 9. f. f (, ) f() = 7 f() = + (, ) (, ) 7 (, ) Relative maimum:, Relative maimum:, Relative minimum:, Relative minimum:, Relative maimum:, Relative maimum:,. Relative minimum:, Answers are the same. f f() = + 7 (, ). Relative minimum:, Answers are the same. f f() = (, ) Houghton Mifflin Compan. All rights reserved. Relative minimum:, Relative maimum:, Relative minimum:, Relative maimum:, Answers are the same. Answers are the same.

29 Section. Graphs of Functions 9. f, <,. f,, >. f,, <. f,, > 7., f,, < > Houghton Mifflin Compan. All rights reserved..., g,, < < 9. f. f,, >. f. 7 h,, 7 f <

30 Chapter Functions and Their Graphs. f. f. f 7. s. g 9. ft t t t t ft ft f is neither even nor odd. Domain:, Domain:, Range:, Range:, Sawtooth pattern Pattern: Sawtooth. f. g. h f. h f is even. g g is odd. h h The function is neither odd nor even.. f. f The function is odd. f. f f The function is neither even nor odd.. Because the domain is s, the function is neither 7. even nor odd.., 7, g s s s g s The function is even. If f is even, another point is,. If f is odd, another point is,. 9., 9 If f is even, another point is Houghton Mifflin Compan. All rights reserved. If f is even, another point is, 7. If f is odd, another point is, 7., 9. If f is odd, another point is, 9.

31 Section. Graphs of Functions 7., 7. If f is even, another point is,. If f is odd, another point is,., If f is even, another point is,. If f is odd, another point is,. 7. a, c 7. If f is even, another point is a, c. If f is odd, another point is a, c. f, even f 9 7. f is neither even 7. f is neither even nor odd. f is even. nor odd h, even 7. f is even. 79. f is neither even nor odd.. gt t is neither even. is neither even nor odd. nor odd. f even nor odd. f. is neither Houghton Mifflin Compan. All rights reserved.., f. f f,

32 Chapter Functions and Their Graphs. f 9. f 9 f or, or,,,, 7. The second model is correct. For instance, C The cost of an -minute -second call is C C $7.9.. Model: Total cost Flat rate Rate per pound C Labels: Total cost C Flat rate 9. Rate per pound., > Equation: C 9.., > Cost of overnight deliver (in dollars) 7 Package weight (in pounds) h top bottom 9., Pt.t.t.t 7.9t 79 t 7 h top bottom, P is increasing from 99 t to 99 t.7, and from t. to. P is decreasing from 99 to. The maimum population was about,, in 99 t.7. Houghton Mifflin Compan. All rights reserved.

33 Section. Graphs of Functions 9. Interval Intake Pipe Drainpipe Drainpipe, Open Closed Closed, Open Open Closed, Closed Closed Closed, Closed Closed Open, Open Open Open, Open Closed Open, Open Open Open, Open Open Closed 9. False. The domain of f is the set of all real numbers. 9. False. The domain must be smmetric about the -ais. 9. c 9. d 97. b 9. e 99. a. f. f a n n a n n... a a f a n n a n n... a a a n n a n n... a a f Therefore, f is odd.. f a n n a n n... a a f a n n a n n... a a a n n a n n... a a f f f ; thus, f is even. Houghton Mifflin Compan. All rights reserved.. f is an even function. g f is even because g f is even because g f f g. g f f f g. g f is even because g f f g.. Yes, defines as. No, does not a function of. (But not as represent as a function of. a function of ) For instance,, and, both lie on the graph. 7. Terms:, Coefficients:, (d). Terms:, 9. Coefficient: g f is neither even nor odd because g f f g nor g.. Answers will var. Terms:,, Coefficients:,,

34 Chapter Functions and Their Graphs. Terms: 7, Coefficient: 7,. d 7 midpoint, 7,. d midpoint,,. d. midpoint,, d 7 7 midpoint,,. f. f 9 f f f f 7 f f f. f f 9 f f 9 f h h h 9 9 h h h 9 f 9 f h f h h h h h h hh h f f f f 9 h, h Houghton Mifflin Compan. All rights reserved.

35 Section. Shifting, Reflecting, and Stretching Graphs. f h h h h h h h h f h f h f f h h h hh h h, h Section. Shifting, Reflecting, and Stretching Graphs You should know the graphs of the most commonl used functions in algebra, and be able to reproduce them on our graphing utilit. Constant function: f c Identit function: f Absolute value function: f (d) Square root function: f (e) Squaring function: f (f ) Cubing function: f You should know how the graph of a function is changed b vertical and horizontal shifts. You should know how the graph of a function is changed b reflection. You should know how the graph of a function is changed b nonrigid transformations, like stretches and shrinks. You should know how the graph of a function is changed b a sequence of transformations. Vocabular Check. quadratic function. absolute value function. rigid transformations. f, f. c >, < c <. ii iv iii (d) i.. h() f (). g() g() f() h() Houghton Mifflin Compan. All rights reserved.. h() g() f() g(). g() f() h() f() h(). g() h() f()

36 Chapter Functions and Their Graphs 7. h () f (). 9. g () f() g () f () h () g() h(). g() f() h(). g () h () f (). f() g() h(). f f f (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (d) f (e) f (f) f (, ) (, ) (, ) (, ) (, ) (, ) (g) Let g f. Then from the graph, g f f g f f g f f g f f (, ) (, ) (, ) (, ) (, ) (, ) 7 (, ) (, ) (, ) (, ) Houghton Mifflin Compan. All rights reserved.

37 Section. Shifting, Reflecting, and Stretching Graphs 7. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (d) (e) (f) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (g) Let g f. Then from the graph, g f f g f f g f f g f f (, ) (, ) (, ( (, (. Horizontal shift three units to left of : (or vertical shift three units upward). Constant function: 7 7. Vertical shift one unit downward of Houghton Mifflin Compan. All rights reserved.. Horizontal shift of : 9. Reflection in the -ais and a vertical shift one unit upward of :. Reflection in the -ais and a vertical shift one unit upward of :. is f reflected in the -ais, followed b a vertical shift one unit downward.. is f shifted verticall. is f shifted right two units. upwards two units.. is f shifted left four units.. is a vertical stretch of f.. is f reflected in the -ais, followed 7. is f shifted left five units. b a horizontal shift to the right three units. f. is shifted down three units. 9. is f reflected in the -ais.. is a vertical stretch of f... is a reflection in the -ais. In fact

38 Chapter Functions and Their Graphs. is a vertical shrink. g f in the -ais followed b a vertical shift upward of four units.. is obtained from b a reflection. g is obtained b a horizontal shift. h is obtained from f b a left of one unit to the right, followed b a reflection in shift of two units and a vertical shrink b a factor the -ais. of.. h is obtained from f b a 7. p is obtained from f b a right shift of one unit, a vertical stretch b a factor horizontal stretch followed b a vertical shift of two, a reflection in the -ais, and a vertical shift two units upward. three units upward.. p is obtained from f b a right shift of two units, followed b a vertical stretch. 9. f g f is a horizontal shift two units to left. h f is a vertical shrink. h g 7 f. f g f is a horizontal shift one unit to the right. h f g h f is a horizontal shrink.. f g f reflection in the -ais and vertical shrink h g f h f reflection in the -ais. f g f is a reflection in the -ais. h f is a horizontal shrink. g h f. f g is obtained from f b a horizontal shift to the left five units, a reflection in the -ais, and a vertical shift upward two units Houghton Mifflin Compan. All rights reserved. (d) g f

39 Section. Shifting, Reflecting, and Stretching Graphs 9. f g is obtained from f b a horizontal shift units to the left, a reflection in the -ais, and a vertical shift units upward. (d) g f. f g is obtained from f b a horizontal shift four units to the right, a vertical stretch of, and a vertical shift upward three units. 7 7 (d) g f. f g is obtained from f b a horizontal shift two units to the left, a vertical shrink of, a reflection in the -ais, and a vertical shift two units downward. 7 7 (d) g f 7. f g is obtained from f b a horizontal shift two units to the right followed b a vertical stretch of. (d) g f Houghton Mifflin Compan. All rights reserved.. f g is obtained from f b a horizontal shift one unit to the left, a vertical shrink, and a reflection in the -ais. (d) g f 9. f g is obtained from f b a horizontal shift one unit to the right, and a vertical shift upward two units. (d) g f

40 Chapter Functions and Their Graphs. f g is obtained from f b a horizontal shift units to the left, a reflection in the -ais, and a vertical shift units downward.. c) f g is obtained from f b a horizontal shift four units to the left, followed b a vertical shift eight units upward.. (d) g f. f g 9 is obtained from f b a horizontal shift three units to the left, followed b a vertical shift nine units upward. (d) g f 9 f g is obtained from f b a horizontal shift two units to the right, a vertical shrink, and a vertical shift three units downward. (d) g f. (d) g f f g is obtained from f b a horizontal shift one unit to the right, a vertical stretch of, a reflection in the -ais, and a vertical shift downward four units. (d) g f. f g is obtained from f b a horizontal shift three units to the left, a vertical shrink, a reflection in the -ais, and a vertical shift one unit downward. (d) g f Houghton Mifflin Compan. All rights reserved.

41 Section. Shifting, Reflecting, and Stretching Graphs. f g is obtained from f b 7. Ft..t is a vertical stretch of f t t, followed b a vertical shift of.. a horizontal shift one unit to the left, a reflection in the -ais, and a vertical shift si units downward. (d) g f Gt Ft..t, t. G F corresponds to 99. G F corresponds to.. Mt.t 79 is a vertical stretch of f t t b., followed b a vertical shift of 79., (d) Mt.t 79 >,.t > t > 9.9 t >.9 The debt will eceed trillion dollars in. Gt Mt.t 79, t G M corresponds to. G M corresponds to 99. Houghton Mifflin Compan. All rights reserved. 9. False. f is a reflection in the -ais.. False. fis a reflection in the -ais.. f is a reflection in the -ais, so the. f is a reflection in the -ais, so the -intercepts are and. -intercepts are and. fis a reflection in the -ais, so the fis a reflection in the -ais, so the -intercepts are and. -intercepts are the same,. f is a vertical stretch, so the f is a vertical stretch, so the -intercepts are the same:,. -intercepts are the same:,. (d) f is a vertical shift, so ou cannot (d) f is a vertical shift, so ou cannot determine the -intercepts. determine the -intercepts. (e) f is a horizontal shift units to the (e) f is a horizontal shift units to the right, so the -intercepts are and. right, so the -intercepts are and.

42 Chapter Functions and Their Graphs. f is a reflection in the -ais, so the. f is a reflection in the -ais, so the graph is increasing on, and graph is increasing on, and decreasing on,. decreasing on, and,. fis a reflection in the -ais, so the fis a reflection in the -ais, so the graph is decreasing on, and increasing graph is increasing on, and decreasing on,. on, and,. f is a vertical stretch, so the f is a vertical stretch, so the graph is increasing on, and decreasing graph is increasing on, and,, on,. and decreasing on,. (d) f is a vertical shift, so the (d) f is a horizontal shift and graph is increasing on, and reflection, so the graph is increasing on, decreasing on,. and decreasing on, and,. (e) f is a horizontal shift one unit to (e) f is a horizontal shift units to the left, so the graph is increasing on, the right, and a vertical shift, so the graph is and decreasing on,. increasing on, and,, and decreasing on,.. The verte is approimatel at, and the graph opens upward. Matches.. The domain is, and, is approimatel on the graph, and f <. Matches. 7. The verte is approimatel, and the graph. The graph of f is shifted to the left opens upward. Matches. approimatel four units, reflected in the -ais, and shifted upward approimatel two units. Matches. 9. Slope L : Slope L : 9 Neither parallel nor perpendicular 7. Slope L : Slope L : 7 7 Neither parallel nor perpendicular 7. Domain: All Domain: 7. f 7 Domain: and 7 f Domain: all real numbers Houghton Mifflin Compan. All rights reserved.

43 Section. Combinations of Functions Section. Combinations of Functions Given two functions, f and g, ou should be able to form the following functions (if defined):. Sum: f g f g. Difference: f g f g. Product: fg fg. Quotient: fg fg, g. Composition of f with g: f g fg. Composition of g with f: g f g f Vocabular Check. addition, subtraction, multiplication, division. composition. g. inner, outer. h. h. 7. Houghton Mifflin Compan. All rights reserved.. h f, g f g f g f g f g fg f g 9 (d) g f f g, Domain: all h

44 Chapter Functions and Their Graphs. f, g f g f g (d) g f Domain: fg 7 7. f, g (d) f g f g f g f g fg f g f g f g, Domain: all.. f, g 9. f g f g 9 fg (d) g f Domain: < < f, g f g f g fg (d) f g Domain: <. f, g (d) f g f g fg Domain: or f g. f, g f g f g fg f g (d), Domain: Houghton Mifflin Compan. All rights reserved. Domain: and or

45 Section. Combinations of Functions. f, g f g (d) g f f g fg Domain:,. f g f g. 9 f g f g f g f g. fg f g. f g f g fg f g 9. g f f g. g f f g. f gt f t gt t t t t 7 Houghton Mifflin Compan. All rights reserved... 7 f gt f t gt. t t t t t t 7t 9 fgt f t gt. t t 9t t fgt f tgt g f f t t g t t t t t t t t t t 7t t t t t t t, t

46 Chapter Functions and Their Graphs. g f f t t g t t t 7. h g f t t, t t. f h g 9 9. f h g. h f g. f, g, f g. f, g g f f g f + g For, f contributes more to the magnitude. For >, g contributes more to the magnitude. f + g f g g contributes more to the magnitude of the sum for. f contributes more to the magnitude of the sum for >.. f, g,. f g f, g, f g 9 9 f both intervals. f f + g g contributes more to the magnitude in f g f + g g contributes more on both intervals. Houghton Mifflin Compan. All rights reserved.

47 Section. Combinations of Functions 7. f, g. f g fg f g f g f g f g f, g f g f g f g f g f g ( f g 7. f, g f g fg f g f g f g f g. f, g ( f g f g f g f gf g ( f g is not defined. Houghton Mifflin Compan. All rights reserved. 9. The domain of f is. Domain of f : or. Domain of g: all real numbers The domain of g is all real numbers. f g f g f Domain of. The domain of f g is all real numbers. f g f :. The domain of f is all real. f, g numbers. Domain of f: The domain of g is all. Domain of g: all f g f g f f g f g f, The domain of f g is. Domain: all

48 Chapter Functions and Their Graphs. The domain of f is all. The domain of g is all real numbers. The domain of is all. f g f. Domain of f: all Domain of g: all Domain of f g f, is all.,. The domain of is all real numbers. f The domain of g is all real numbers. f g f g f Domain: all real numbers. f Domain of f: all Domain of g: all, g 7. The domain of f is all real numbers. f g f g f Domain: all The domain of f g f g f Domain: ± g is all ±. Domain of f: all ± Domain of g: all real numbers Domain of f g f. is all real numbers,. f g fg f Domain: all g f g f g The are equal. f g g f f g = g f 9.. f g fg f Domain: all g f g f g g f, f g f g fg f 9 Domain: all g f g f g The domain of f g = g f The are not equal. f g is all real numbers. The are equal. Houghton Mifflin Compan. All rights reserved.

49 Section. Combinations of Functions 9. f g g f. f g fg f Domain: Domain: all The are equal. g f g f g f g = g f The are equal. f g = g f. f g fg f Domain: all g f g f g g f f g f g g f. f g f g f g f g f g No, f g g f because. f g g f 9 9. f g f g f g The are equal because. f g g f Houghton Mifflin Compan. All rights reserved. 7.. f g f g f g f g f g, No, f g g f because. f g f g f g The are not equal because. f g g f f g 7 g f 7 9

50 Chapter Functions and Their Graphs 9. f g f g f g f g f g No, f g g f because. f g g f 7. f g fg f g f g The are not equal because. f g g f 7. f g f g f f g g. f g fg f g f g f g. f g f g fg f g. f g f g f g f g f g. Let f and g, then f g h. This is not a unique solution. For eample, if f and g, then f g h as well... h 7. Let f and g, then One possibilit: Let g and f. f g h. This answer is not unique. Other possibilities ma be: f g f h f and g or f and g or f 9 and g h 9 9. Let f and g, then One possibilit: Let g 9 and f. f g h. Again, this is not a unique solution. Other possibilities ma be: f g f 9 9 h f and g or f and g Houghton Mifflin Compan. All rights reserved.

51 Section. Combinations of Functions 7. h 7. Let f and g. Then f g h. (Answer is not unique.) One possibilit: Let g and f. f g f 7. h One possibilit: Let g and f. f g f g f h 7. T R B T B R B contributes more to T at higher speeds. 7. R R R t.t.7t 7 7.t.t, t,,,,,, R R R 7. t corresponds to 99. Houghton Mifflin Compan. All rights reserved. 7. Year A rt gives the area of the circle as a T function of time. T represents the total out-of-pocket paments, insurance premiums and other tpes of premiums in billions of dollars. A rt Art A.t.t.t

52 Chapter Functions and Their Graphs 7. r Ar r A r Ar A A r represents the area of the circular base of the tank with edge. 79. C 7 t t Ct Ct t 7 t 7 Ct represents the cost after t hours. units,, t.7, or hours minutes. miles ( mph)( t hours). N Tt NTt. Area r, rt.t. Hence Nt A rt.t 7.t, t t t A r t 9 7 square meters N T represents the number of bacteria as a function of time. N T At time t, there are bacteria. A 7. t t 7. hours N when t. hours.. gf g,., represents percent of the amount over $,.. miles ( mph)( t hours) s t t t t R p S.9p R Sp.9p S Rp.9p (d) R S,,7 S R,, The discount first ields a lower cost.. False. f g f, but. True. f g f g is onl defined if g is in g f g. the domain of f. Houghton Mifflin Compan. All rights reserved.

53 Section. Combinations of Functions 7. Let A, B, and C be the three siblings, in decreasing. From Eercise 7, A B and B age. Then and B C C. A B. B C and B A B A. Hence, C C C A A. If A, then B and C. If C, then B 7 and A. 9. Let f and g be odd functions, and define h f g. Then, h fg fg fg h. since f and g are both odd Thus, h is even. Let f and g be even functions, and define h f g. Then, h fg fg h. Thus, h is even. since f and g are both even 9. The product of an odd function and an even function is odd. Let f be odd and g even. Then fg f g f g fg Thus, fg is odd. 9. g f f f f g, which shows that g is even. h f f f f f f h, which shows that h is odd. Houghton Mifflin Compan. All rights reserved. 9. f f f f f where g is even and h is odd. f g g h 9.,,,,, 7 other answers possible 9. Three points on the graph of are,,,. and,..

54 Chapter Functions and Their Graphs 9.,,,,, 9. Three points on the graph of are other answers possible,,, and, Section. Inverse Functions. Two functions f and g are inverses of each other if fg for ever in the domain of g and g f for ever in the domain of f. Be able to find the inverse of a function, if it eists.. Replace f with.. Interchange and.. Solve for. If this equation represents as a function of, then ou have found f. If this equation does not represent as a function of, then f does not have an inverse function. A function f has an inverse function if and onl if no horizontal line crosses the graph of f at more than one point. A function f has an inverse function if and onl if f is one-to-one. Vocabular Check. inverse, f. range, domain.. one-to-one. Horizontal f. f f f f f f f f f f f f f f f Houghton Mifflin Compan. All rights reserved.

55 Section. Inverse Functions. f 7. f 7 f f f f f f f f f f f f f f. f f f f f f f. f f f f f 7. f f f f f f f. f f f f f f f f Houghton Mifflin Compan. All rights reserved. 9.. f g f g f g f g Note that the entries in the tables are the same ecept that the rows are interchanged. f g f 9 g f g f g 9 7 The entries are the same ecept that the rows are interchanged.

56 Chapter Functions and Their Graphs. f g f g f g f g Note that the entries in the tables are the same ecept that the rows are interchanged.. f g f g f g f g The entries are the same ecept that the rows are interchanged.. f g f Since, g f g 9 7 f g 9 7. Note that the entries in the tables are the same ecept that the rows are interchanged. f g f g f g f g The entries in the table are the same ecept that the rows are interchanged. Houghton Mifflin Compan. All rights reserved.

57 Section. Inverse Functions 7. fg f. g f g f g f, g f g f g f g Reflections in the line f = g Reflections in the line 7. fg f, g f g g f Reflections in the line. f 9, g 9, 9 fg f g f g9 9 9 f g Reflections in the line 9. fg f g f g f g Houghton Mifflin Compan. All rights reserved.. f, ; g, < f g f g f g Reflections in the line g f Reflections in the line. The inverse is a line through,.. The inverse is a line through, and,. Matches graph. Matches graph.

58 Chapter Functions and Their Graphs. The inverse is half a parabola starting at,.. The inverse is a reflection in of a third-degree Matches graph. equation through,. Matches graph (d).. f, g f g f Reflection in the line g The entries in the tables are the same, ecept that the rows are interchanged.. f g g f 9 9 f g The graphs are reflections in the line. The entries in the table are the same ecept that the rows are interchanged. 7.. f, f f g g g Reflection in the line f g g f f g f g The entries in the tables are the same, ecept that the rows are interchanged. f The entries in the table are the same ecept that the rows are interchanged. 9 9 g Houghton Mifflin Compan. All rights reserved. Reflection in the line.

59 Section. Inverse Functions 9 9. Not a function. It is the graph of a function, but not one-to-one.. It is the graph of a one-to-one function.. It is the graph of a one-to-one function.. It is the graph of a one-to-one function.. It is the graph of a one-to-one function.. f. f is one-to-one because a horizontal line will intersect the graph at most once. f 7. h h is not one-to-one because some horizontal lines intersect the graph twice. f does not pass the Horizontal Line Test, so f is not one-to-one.. g 9. h. f h is not one-to-one because some horizontal lines intersect the graph twice. is not one-to-one because it does not pass the Horizontal Line Test. g does not pass the Horizontal Line Test, so g is not one-toone. Houghton Mifflin Compan. All rights reserved.. f. f.. f is not one-to-one because the horizontal line intersects the graph at ever point on the graph. is not one-to-one because it does not pass the Horizontal Line Test.. f 7. is one-to-one because it passes the Horizontal Line Test. h is not one-to-one because some horizontal lines intersect the graph more than once. h. g g is one-to-one because a horizontal line will intersect the graph at most once. is not one-to-one because it does not pass the Horizontal Line Test. f

60 Chapter Functions and Their Graphs 7. f ± f is not one-to-one. This does not represent as a function of. f does not have an inverse.. g is not one-to-one. For eample, g g. 9. f 9 9. f f is one-to-one. f f f is one-to-one and has an inverse.. f is not one-to-one, and does not have an. h is not one-to-one. inverse. For eample, f f. For eample, h h.. f,,. q, is one-to-one.,,,,,,,, f is one-to-one. This is a function of, so f has an inverse. f, 9,,, The inverse is q. Houghton Mifflin Compan. All rights reserved.

61 Section. Inverse Functions. f,.,,,,,,,, f is one to one. This is a function of, so f has an inverse. f, f,,,,,,,,, f is one-to-one, so f has an inverse. f, 9 7. f,,,, 9 9 since.,, f, Houghton Mifflin Compan. All rights reserved.. f f is not one-to-one. For instance f f. Hence, f does not have an inverse.. f f Reflections in the line 9. f. f f f f f f Reflections in the line f f Reflections in the line f

62 Chapter Functions and Their Graphs. f f f. f f f Reflections in the line f Reflections in the line f. f, f f. f, f Reflections in the line f = f Reflections in the line f,. f, 7. f,, f = f f f f Reflections in the line. f, > f, > f f 9. If we let f,, then f has an inverse. Note: We could also let. f,,,,,,,,,, Thus, f,. Houghton Mifflin Compan. All rights reserved.

63 Section. Inverse Functions 7. If we let f,, then f has an 7. If we let then f has an inverse. [Note: We could also let. ] inverse. Note: We could also let. f,,,,,,,,, Thus, f,. f,, f, f when,,,,,, Thus, f,. 7. If we let f,, then f has an inverse. [Note: We could also let ] f,. f when.,,,,,, Thus, f,. 7. Let f,. f Domain f : Domain f : Range f : Range f : Houghton Mifflin Compan. All rights reserved. 7. Let f,. f Domain f : Range f : Domain f : Range f : 7. Let f,. f Domain f : Range f : Domain f : Range f : 7. Let f,. 77. Let f, and. f Domain f : Range f : Domain f : Range f : because. f, Domain f : Range f : Domain f : Range f :

64 Chapter Functions and Their Graphs 7. Let f, and. because. f, Domain f : Range f : Domain f : Range f : 79. f f. f f. f because f.. g because g.. f g f. g f g. f g f. g f g 7. g f g. f g f 9. f The graph of the inverse relation is an inverse function since it satisfies the Vertical Line Test. f f 9. and h h Not an inverse function since it does not satisf the Vertical Line Test. Houghton Mifflin Compan. All rights reserved.

65 Section. Inverse Functions 9. g g 9. and The graph of the inverse relation is not an inverse function since it does not satisf the Vertical Line Test. g f f Inverse function since it satisfies the Vertical Line Test. In Eercises 9 9, f, f, g, g f g f g f g f g f g g 9. f f f f f f g g g g g fg fg f 9. Now find the inverse of f g : f g Note: f g g f g f g f g In Eercises 99, f, f, g, g. Houghton Mifflin Compan. All rights reserved. 99. g f g f g. f g f g f

66 Chapter Functions and Their Graphs. f g fg f. Now find the inverse of f g : f g Note that f g g f ; see Eercise 99.. g f g f g. Now find inverse: g f Note that g f f g.. Yes, f is one-to-one. For each European shoe size, there is eactl one U.S. shoe size. f f because f. (d) f f f (e) f f f 7. Yes, g is one-to-one. For each European shoe size, there is eactl one U.S. shoe size. g g 9 because g9. (d) gg 9 g7 9 (e) g g g 7. Yes, f is one-to-one, so f eists. f gives the ear corresponding to the values in the second column. f. because f.. (d) No, because f f f.7 hourl wage number of units produced If units are produced, then (d) If the hourl wage is $., then.7 $ units. Houghton Mifflin Compan. All rights reserved.

67 Section. Inverse Functions 7 7. False. f is even, but f does not eist.. True. If, b is the -intercept of f, then b, is the -intercept of f. 9. We will show that f g g f for all in their domains. Let f g f g then f g f g. Hence, g f g f Thus, g f f g. g g f g.. If f is one-to-one, then f eists. If f is odd, then f f. Consider f f. Then f f f f f f. Thus, f is odd.. No, the graphs are not reflections of each other in the line.. Yes, the graphs are reflections of each other in the line.. Yes, the graphs are reflections of each other in the line.. Yes. The inverse would give the time it took to complete n miles.. Yes, the graphs are reflections of each other in the line.. Yes, assuming that the population is increasing between 9 and. The inverse would give the ear corresponding to a given population. 7. No. The function oscillates.. No, because heights remain constant, or even decrease, after man ears ,..,.,. Yes, is a function of., Houghton Mifflin Compan. All rights reserved... No. Does not pass. 9. Vertical Line Test ±9 7. Yes, is a function of. No, is not a function of.. ± No, is not a function of. Yes, is a function of.

68 Chapter Functions and Their Graphs Section.7 Linear Models and Scatter Plots You should know how to construct a scatter plot for a set of data You should recognize if a set of data has a positive correlation, negative correlation, or neither. You should be able to fit a line to data using the point-slope formula. You should be able to use the regression feature of a graphing utilit to find a linear model for a set of data. You should be able to find and interpret the correlation coefficient of a linear model. Vocabular Check. positive. negative. fitting a line to data.,.. Monthl sales (in thousands of dollars) Years of eperience Score on second quiz Score on first quiz Yes, the data appears somewhat linear. The more eperience,, corresponds to higher sales,. No. Quiz scores are dependent on several variables, such as stud time, class attendance, etc.. Negative correlation. No correlation. No correlation. Positive correlation decreases as increases. 7. (, ) (, ) (, ).. Correlation coefficient:.99 = + (, ) (, ). (, ) = + (, ) (, ) (, ) (, ).. Correlation coefficient: Houghton Mifflin Compan. All rights reserved. (d) Yes, the model appears valid. (d) The model appears valid.

69 Section.7 Linear Models and Scatter Plots (, ) (, ) (, ) (, ) (, ) = 7 = + 7. (, ) (, 7) (, ) (, ) (, ) Correlation coefficient: Correlation coefficient:.97 9 (d) Yes, the model appears valid. (d) The model is somewhat valid. Houghton Mifflin Compan. All rights reserved... Elongation 7 d Force d.7f. d.f or F.d.9 (d) If F, d.. cm..t F..t. Yes, the model is a good fit. (d) For, t and.7 minutes. For, t and.7 minutes. Yes, the answers seem reasonable. (d) For, t and., or $,,. For, t and 97, or $,97,. Yes, the answers seem reasonable. (e) The slope is.. It sas that the mean salar increases b $, per ear. Yes, the model is a good fit.

70 7 Chapter Functions and Their Graphs...t.9 C.t.7 Correlation coefficient:.99 Yes, the model is a good fit. (d) For, t and., or $,. For, t and.7, or $,7. Yes, the answers seem reasonable. (d) The model is a good fit. (e) For, t, $.9. For, t, $.7. (f) Answers will var.., 7. 7 P.t P.t, 7. The model is a good fit. (d) For, t and P,, or,, people. Answers will var Correlation coefficient:. The slope represents the increase in sales due to increased advertising. (d) For $,. and 7. or $7,. The model is not a good fit. (d) For, t and P, or, people. Answers will var. Houghton Mifflin Compan. All rights reserved.

71 Section.7 Linear Models and Scatter Plots 7 9. T.7t 9 Correlation coefficient:.799 The slope indicated the number of new stores opened per ear. (d) T.7t 9 >.7t > 7 t >. The number of stores will eceed near the end of. (e) Year Data 7 7 Model 9 77 The model is not a good fit, especiall around t.. The negative slope indicates that the times are decreasing..t. (d) The model is not ver accurate. (e) Answers will var.. True. To have positive correlation, the -values tend to increase as increases.. False. The closer to or, the better the fit.. Answers will var.. Answers will var. Houghton Mifflin Compan. All rights reserved f. g f fw w w g gz z z h,, w w 7 > h h , z z. k k ± 9 7 ± 7 ± ±,

72 7 Chapter Functions and Their Graphs Review Eercises for Chapter... m (, ) (, ). Slope, undefined. 7 7 m 7 7 (7, ) (, (, (7, ) ( (. Slope 7..,,., m.... ), ) Slope 7 (.7,.) ), )...7 (,.) (., ) (., ) Three additional points:,,,,,, (other answers possible) Houghton Mifflin Compan. All rights reserved.

73 Review Eercises for Chapter 7. 9 Three additional points:,,,,,, (other answers possible). Three additional points:,,,,,, (other answers possible).. Three additional points:,,,,,, (other answers possible) Three additional points:,,,, 7, 7, (other answers possible) Houghton Mifflin Compan. All rights reserved... 7 Three additional points:, 7,,, 7, 7, 9 (other answers possible) horizontal line Three additional points:,,,,, (other answers possible). Three additional points:,,,,, (other answers possible) 7. m is undefined means that the line is vertical.. Slope is undefined, line is vertical: or Three additional points:,,,,, (other answers possible) Three additional points:,,,,, (other answers possible)

74 7 Chapter Functions and Their Graphs 9.. Slope is undefined. Line is vertical. Slope t corresponds to.. Point:,,, slope: V, t V t 7 m Point:, 79 V 79 t V t 7. m.7. t corresponds to. Point:,. Point:, 7.9, slope:. V..7t V 7.9.t V.7t.9 V.t. 7.,,,,,. Point:,, slope:.7 m,, S,,t, S,t, For the fourth quarter let t. Then we have S,, $,. V.7t V.7t. In, t and V 7 dollars. (d) V when t.,. Algebraicall, V.7t. t....7 Houghton Mifflin Compan. All rights reserved.

75 Review Eercises for Chapter 7 9. and m Parallel slope: m Perpendicular slope: m. Slope of given line: m is a vertical line; the slope is not defined.. is a horizontal line. Parallel line: Parallel line through, : Perpendicular slope: m Perpendicular line through,: Perpendicular line: 7 Houghton Mifflin Compan. All rights reserved.. Not a function. is assigned two different values. Function Function (d) Not a function. No value is assigned to.. No, is not a function of. Some -values correspond to two -values. For eample, corresponds to and. 7. Each value,, corresponds to onl one -value so is a function of.. Not a function. u is assigned two different values. Function Function (d) Not a function. w is assigned two different values and u is unassigned.. Yes,.. No, does not pass Vertical Line Test.