2.4 COMPOSITE FUNCTIONS INVERSE FUNCTIONS PIECEWISE FUNCTIONS

Transcription

1 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 011, Connall.4 COMPOSITE FUNCTIONS INVERSE FUNCTIONS PIECEWISE FUNCTIONS

2 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 011, Connall 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 b using the output o the unction g as the input to. also ( g( )) ( g)( )

3 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 011, Connall Composition o Functions Eample 3 (a) Let () = + 1 and g() = 3. (a) Calculate (g(3)) and g((3))

4 Composition o Functions Solution (a) g(3) = (3) 3 = 6, so (g(3)) = (6) (6) = (6) + 1 = 13, so (g(3)) = 13 To calculate g((3)), we have (3) = (3)+1=7 g((3)) = g(7) g(7) = (7) 3 = 46, so g((3)) = 46 Note in this case, (g(3)) g((3)). Functions Modeling Change: A Preparation or Calculus, 4th Edition, 011, Connall

5 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 011, Connall Composition o Functions Solution (b) In general, the unctions (g()) and g(()) are dierent: (g()) = ( 3) = ( 3) + 1 = = 5 g(()) = g( + 1) = ( + 1) 3 = = 4 + 4

6 A circular oil slick is epanding with radius, r in ards, at time t in hours given b r t 0.1t, or t in hours, 0< t < 10. Find a ormula or the area in square ards, A = (t), as a unction o time.

7 r t 0.1t substitute A r A ( t 0.1t ) then simpli A (t 0.1t )(t 0.1t ) oil A (4t 0.t 0.t 0.01t 4 ) A (4t 0.4t 0.01t 4 )

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

9 R( (p)) price /barrel # barrels o oil revenue earned so, revenue (price)

10 )) ( ( )), ( ( (1)), ( (0)), ( : 3 ) ( 1, ) ( : g g g g ind g given ) ( ) (

11 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 011, Connall 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 NOT AN EXPONENT 1 Inverse Function Procedure Reassign the variables, then solve or

12 ( 0)? (?) 0 3 INVERSES 1 (0)? 1 (?) 0 3

13 (10.)Generate inverse Omit:17-0, 5, 6.

14 # reassign variables Inverse 3 mabe a unction 1

15 #16 Inverse mabe a unction reassign 1 ) (1 1) ( 1 1 1

16 # reassign (4 ) 4 7

17 #3 continued ) 7 (

18 #4 Generate the inverse 3 11

19 Generate the inverse

20 Generate the inverse 5 4 9

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

22 0, (0), 1 (0), 3, (3), 1 (3) 1 (3) (3) 0 (0) 1 (0) 3

23 #81 Use Figure (a) Evaluate (g (a)). (b) Evaluate g ( (c)). (c) Evaluate (b) g (b). (d) For what positive value(s) o is () g ()?

24 (a) Evaluate (g (a)) a

25 (b) Evaluate g ( (c)) b

26 1 1 (c) Evaluate ( b) g ( b) 0 ( c) c

27 (d) For what positive value(s) o is () g ()? a

28 HW: A compan 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 quantit demanded was 500 units per week. When the unit price was raised to $4, the quantit demanded dropped to 300 units per week. Let D(p) be the quantit per week demanded b consumers at a unit price o $p.

29 (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) Currentl, the compan can produce 400 units ever week. What should the price o the product be i the compan wants to sell all 400 units? () I the compan produced 500 units per week instead o 400 units per week, would its weekl revenues increase, and i so, b how much?

30 The predicted pulse in beats per minute (bpm) o a health person iteen minutes ater consuming q milligrams o caeine is given b 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-toic levels o caeine. What do each o the ollowing statements tell ou about caeine and a person's pulse?

31 )) ( (1.1 6) 0) ( 5) (0) ) ( 4) 0 ) ( 3) 0) ( ) ) ( 1) c 1 c c 1 c c 1 c 1 c q q r q r r q

32 Functions Modeling Change: A Preparation or Calculus, 4th Edition, 011, Connall.3 PIECEWISE DEFINED FUNCTIONS

33 KNOW BASIC GRAPHS

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47 Without a calculator, sketch all our unctions on the same ais and label each along with the coordinates o all intercepts and intersecting points ( ) 4 g( ) 5 h( ) 4 j( ) 5