COMPOSITE AND INVERSE FUNCTIONS & PIECEWISE FUNCTIONS

Transcription

1 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally 2.4 COMPOSITE AND INVERSE FUNCTIONS & PIECEWISE FUNCTIONS

2 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally Composition o Functions For two unctions (t) and g(t), the unction (g(t)) is said to be a composition o with g. The unction (g(t)) is deined by using the output o the unction g as the input to. also ( g( x)) ( g)( x)

3 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally Composition o Functions Solution (a) g(3) = = 6, so (g(3)) = (6). (6) = = 13, so (g(3)) = 13. To calculate g((3)), we have g((3)) = g(7), because (3) = 7 Then g(7) = = 46, so g((3)) = 46 Notice that, (g(3)) g((3)).

4 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally Composition o Functions Solution (b) In general, the unctions (g(x)) and g((x)) are dierent: (g(x)) = (x 2 3) = 2(x 2 3) + 1 = 2x = 2x 2 5 g((x)) = g(2x + 1) = (2x + 1) 2 3 = 4x 2 + 4x = 4x 2 + 4x 2

5 A circular oil slick is expanding with radius, r in yards, at time t in hours given by r 2t 0.1t 2, or t in hours, 0< t < 10. Find a ormula or the area in square yards, A = (t), as a unction o time.

6 Give the meaning and units o the composite unction R( (p)), where Q = (p) is the number o barrels o oil sold by a company when the price is p dollars/barrel and R( Q) is the revenue earned in millions o dollars.

7 given : ( x) x 2 1, g( x) 2x 3 ind : ( g(0)), ( g(1)), ( g( x)), g( ( x))

8 (0)? (?) 0 1 (0)? 1 (?) 0

9 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally Inverse Functions The roles o a unction s input and output can sometimes be reversed. The unctions and g are called inverses o each other. A unction which has an inverse is said to be invertible. Inverse Function Notation 1 Inverse Function Procedure Reassign the variables then solve or y

10 Generate inverse Omit:17-20, 25, 26.

11 Figure 8.18 deines the unction. Rank the ollowing quantities in order rom least to greatest:

12 0, (0), 1 (0), 3, (3), 1 (3)

13 A company believes there is a linear relationship between the consumer demand or its products and the price charged. When the price was $3 per unit, the quantity demanded was 500 units per week. When the unit price was raised to $4, the quantity demanded dropped to 300 units per week. Let D(p) be the quantity per week demanded by consumers at a unit price o $p.

14 (a) Estimate and interpret D(5). (b) Find a ormula or D(p) in terms o p. (c) Calculate and interpret D -1 (5). (d) Give an interpretation o the slope o D(p) in terms o demand. (e) Currently, the company can produce 400 units every week. What should the price o the product be i the company wants to sell all 400 units? () I the company produced 500 units per week instead o 400 units per week, would its weekly revenues increase, and i so, by how much?

15 The predicted pulse in beats per minute (bpm) o a healthy person iteen minutes ater consuming q milligrams o caeine is given by r = (q). The amount o caeine in a serving o coee is q c and r c = (q c ). Assume that is an increasing unction or non-toxic levels o caeine. What do each o the ollowing statements tell you about caeine and a person's pulse?

16 )) ( (1.1 20) ( (0) ) ( 20 ) ( 2 20) ( ) ( c c c c c c c q q r q r r q

17 KNOW BASIC GRAPHS

18 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally Example A Piecewise Deined Function A unction may employ dierent ormulas on dierent parts o its domain. Such a unction is said to be piecewise deined. For example, the unction graphed has the ollowing ormulas: y = x 2 y y = 6 - x y 2 x 6 x or or x x x

19 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally The Absolute Value Function The Absolute Value Function is deined by ( x) x x x or or x x 0 0

20 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally Graph o y = x (-3,3) y (3,3) (-2,2) (2,2) (-1,1) (1,1) (0,0) x The Domain is all real numbers, The Range is all real nonnegative numbers.

21 Text Example 3 The Ironman Triathlon is a race that consists o three parts: a 2.4 mile swim ollowed by a 112 mile bike race and then a 26.2 mile marathon. A participant swims steadily at 2 mph, cycles steadily at 20 mph, and then runs steadily at 9 mph. Assuming that no time is lost during the transition rom one stage to the next, ind a ormula or the distance d, covered in miles, as a unction o the elapsed time t in hours, rom the beginning o the race. Graph the unction.

22 d 2t 20t 21.6 or or t 1.2 x 6.8 9t 53.2 or 6.8 x 9.71

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24 Where do the equations and the end points come rom? d 2t 20t 21.6 or or t 1.2 x 6.8 9t 53.2 or 6.8 x 9.71

25 The charge or a taxi ride in New York City is $2.50 or the irst 1/4 o a mile, and $0.40 or each additional ¼ o a mile (rounded up to the nearest 1/4 mile). (a) Make a table showing the cost o a trip as a unction o its length. Your table should start at zero and increase to two miles in 1/4-mile intervals.

26 (b) What is the cost or a 1.25-mile trip? (c) How ar can you go or $5.30? (d) Graph the cost unction in part (a).

27 Graph the piecewise unction ( x) 2x x 2 4 or or 9 x 3 3 x 5 x 1 2 or 7 x 100

28 88-21 The rate at which people enter an oice building is given in the graph ollowing. A negative rate means that people are leaving the building.

29 RATE 0 A B C D E F G H I J TIME K

30 1) Write appropriate units or each o the variables.

31 2) Create reasonable coordinate values or the 12 blue points indicated on the graph.

32 For each o the ollowing statements give the largest interval on which:

33 (a) The number o people in the building is increasing.

34 (b) The number o people in the building is constant.

35 (c) The number o people in the building is increasing astest.

36 (d) The number o people in the building is decreasing.

37 Using the values you created, generate ormulas or each o the 10 straight lines o the graph.

38 Suppose w = j (x) is the average daily quantity o water (in gallons) required by an oak tree o height x eet. (a) What does the expression j(25) represent? What about j -l (25)?

39 (b) What does the ollowing equation tell you about v: j (v) = 50. Rewrite this statement in terms o j -l. (c) Oak trees are on average z eet high and a tree o average height requires p gallons o water. Represent this act in terms o j and then in terms o j -l. (d) Using the deinitions o z and p rom part (c), what do the ollowing expressions represent?

40 j(2z) 2 j( z) j( z 10) j( z) 10 j 1 (2 p) j 1 ( z 10) j 1 ( z) 10

41 Inverse Functions The roles o a unction s input and output can sometimes be reversed. For example, the population, P, o birds on an island is given, in thousands, by P = (t), where t is the number o years since In this unction, t is the input and P is the output. I the population is increasing, knowing the population enables us to calculate the year. Thus we can deine a new unction, t = g(p), which tells us the value o t given the value o P instead o the other way round. For this unction, P is the input and t is the output. The unctions and g are called inverses o each other. A unction which has an inverse is said to be invertible. Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally

42 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally Inverse Function Notation I we want to emphasize that g is the inverse o, we call it 1 (read -inverse ). To express the act that the population o birds, P, is a unction o time, t, we write P = (t). To express the act that the time t is also determined by P, so that t is a unction o P, we write t = 1 (P). The symbol 1 is used to represent the unction that gives the output t or a given input P. Warning: The 1 which appears in the symbol 1 or the inverse unction is not an exponent.