Functions Modeling Change A Preparation for Calculus Third Edition

Transcription

1 Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1

2 Section 2.4 Composition and Inverse Functions 2

3 Composition and Inverse Functions Two functions may be connected by the fact that the output of one is the input of the other. Page 79 3

4 Let's define a new function: Cost (C) as a function of # of gallons of paint (n): C g( n) $30.50n Page 79 4

5 Cost (C) as a function of # of gallons of paint (n): C g( n) $30.50n Previously, we saw- # of gallons of paint (n) as a function of house Area (A): n f ( A) A 250 Page 79 Example #1 5

6 Now we want Cost (C) as a function of house Area (A): C g( n) $30.50n n f ( A) A 250 Page 79 6

7 C g( n) $30.50n n f ( A) A 250 Now we want Cost (C) as a function of house Area (A): C g( n) & n f ( A) C h( A) g( f ( A)) Page 79 7

8 C g( n) $30.50n n f ( A) A 250 C g( n) & n f ( A) C h( A) g( f ( A)) Page 79 A A 8

9 C g( n) $30.50n n f ( A) A 250 C g( n) & n f ( A) C h( A) g( f ( A)) A h = composition of functions f & g A Page 79 f = inside function, g = outside function 9

10 You will recall (!) Temperature (T) as a function of chirp Rate (R): T 1 4 R 40 Page 79 Example 2 10

11 You will recall (!) Temperature (T) as a function of chirp Rate (R): T 1 4 R 40 Let's define a new function- Chirp Rate (R) as a function of time (x): R g( x) 20 x 2 Here, x is in hrs. since midnight & 0 x 10 Page 79 11

12 Now we want Temperature (T) as a function of time (x): T 1 4 R 40 R g( x) 20 x 2 Page 79 12

13 T 1 4 R 40 R g( x) 20 x Now we want Temperature (T) as a function of time (x): T f ( R) & R g( x) T h( x) f ( g( x)) 2 Page 79 13

14 1 T R 40 4 R g( x) 20 x 2 T f ( R) & R g( x) T h( x) f ( g( x)) 1 (20 x 2 ) (0 x 10) x 2 Page 79 14

15 T 1 R 4 40 R g( x) 20 x T f ( R) & R g( x) T h( x) f ( g( x)) 2 1 (20 2 x ) (0 x 10) h = composition of functions f & g x 2 Page 79 f = outside function, g = inside function 15

16 If f(x) = x 2 and g(x) = 2x + 1, find (a) f(g(x)) (b) g(f(x)) Page 80 Example #3 16

17 If f(x) = x 2 and g(x) = 2x + 1, find (a) f(g(x)) Page 80 17

18 If f(x) = x 2 and g(x) = 2x + 1, find (a) f(g(x)) f ( g( x)) f (2x 1) 18

19 If f(x) = x 2 and g(x) = 2x + 1, find (a) f(g(x)) f ( g( x)) f (2x 1) (2x 1) 2 Page 80 19

20 If f(x) = x 2 and g(x) = 2x + 1, find (a) f(g(x)) f ( g( x)) f (2x 1) (2x 1) 2 2 4x 4x 1 Page 80 20

21 If f(x) = x 2 and g(x) = 2x + 1, find (b) g(f(x)) Page 80 21

22 If f(x) = x 2 and g(x) = 2x + 1, find (b) g(f(x)) g( f ( x)) g( x ) 2 22

23 If f(x) = x 2 and g(x) = 2x + 1, find (b) g(f(x)) g( f ( x)) g( x ) 2 2 2( x ) 1 Page 80 23

24 If f(x) = x 2 and g(x) = 2x + 1, find (b) g(f(x)) g( f ( x)) g( x ) ( x ) 1 2x 1 Page 80 24

25 If f(x) = x 2 and g(x) = 2x + 1, find (a) f(g(x)) f ( g( x)) f (2x 1) (2x 1) 2 2 4x 4x 1 Page 80 25

26 If f(x) = x 2 and g(x) = 2x + 1, find (b) g(f(x)) g( f ( x)) g( x ) ( x ) 1 2x 1 Page 80 26

27 Inverse Functions Page 80 27

28 Inverse Functions The roles of a function's input and output can sometimes be reversed. Page 80 28

29 Inverse Functions Example: the population, P, of birds is given, in thousands, by P = f(t), where t is the number of years since (Here t = input, P = output.) Define a new function, t = g(p), which tells us the value of t given the value of P instead of the other way round. (Here, P = input, t = output.) The functions f and g are called inverses of each other. A function which has an inverse is said to be invertible. Page 80 Example #4 29

30 Inverse Function Notation f-inverse: f 1 (not an exponent!) Back to our example: P = f(t) original function t = g(p) = f 1 (P) inverse function Page 80 30

31 Back to our example: Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007): (a) What does f(4) represent? (b) What does f 1 (4) represent? Page 80 31

32 Back to our example: Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007): (a) What does f(4) represent? Page 80 32

33 Back to our example: Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007): (a) What does f(4) represent? Bird population in the year = Page 80 33

34 Back to our example: Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007): (b) What does f 1 (4) represent? Page 80 34

35 Back to our example: Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007): (b) What does f 1 (4) represent? t = g(p) = f 1 (P) Page 80 35

36 Back to our example: Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007): (b) What does f 1 (4) represent? t = g(p) = f 1 (P) Population = input, time = output Page 80 36

37 Back to our example: Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007): (b) What does f 1 (4) represent? t = g(p) = f 1 (P) t = g(4) = f 1 (4) Population = input, time = output Page 80 37

38 Back to our example: Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007): (b) What does f 1 (4) represent? t = g(p) = f 1 (P) t = g(4) = f 1 (4) f 1 (4) = # of years (since 2007) at which there were 4,000 birds on the island. Page 80 38

39 You will recall (!!) Temperature (T) as a function of chirp Rate (R): 1 T f ( R) R 40 4 What is the formula for the inverse function, R= f 1 (T)? Page 81 Example 5 39

40 You will recall (!!) Temperature (T) as a function of chirp Rate (R): 1 T f ( R) R 40 4 What is the formula for the inverse function, R= f 1 (T)? Solve for R... Page 81 40

41 What is the formula for the inverse function, R= f 1 (T)? T Solve for R... T 40 4( T 40) R R R 40 Page 81 4T 160 R f ( T ) 1 41

42 Domain & Range of an Inverse Function Page 81 42

43 Domain & Range of an Inverse Function The input values of the inverse function f 1 are the output values of the function f. Page 81 43

44 Domain & Range of an Inverse Function The input values of the inverse function f 1 are the output values of the function f. Therefore, the domain of f 1 is the range of f. Page 81 44

45 What about the domain & range of the cricket function T=f(R) and the inverse R= f 1 (T)? 1 T f ( R) R R f ( T ) 4T 160 Page 81 45

46 1 T f ( R) R R f ( T ) 4T 160 Page 81 46

47 1 T f ( R) R R f ( T ) 4T T f ( R) R 40 4 For if a realistic domain is 0 R 160, then the range of f is 40 T 80. Page 81 47

48 A Function and its Inverse Undo Each Other Page 81 48

49 A Function and its Inverse Undo Each Other Calculate the composite functions: f 1 (f(r)) & f(f 1 (T)) for the cricket example. Interpret the results. Page 81 Example #6 49

50 1 T f ( R) R R f ( T ) 4T f 1 ( f ( R)) 4 R R R Page 81 50

51 1 T f ( R) R R f ( T ) 4T f ( f 1 ( T )) 4T T T Page 81 51

52 The functions f and f 1 are called inverses because they undo each other when composed. Page 81 52

53 End of Section