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2. Section 2.4 Composition and Inverse Functions 2

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

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

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

6. Now we want Cost (C) as a function of house Area (A): 6 Page 79

7. Now we want Cost (C) as a function of house Area (A): 7 Page 79

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9. h = composition of functions f & g f = inside function, g = outside function 9 Page 79

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

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

12. Now we want Temperature (T) as a function of time (x): 12 Page 79

13. Now we want Temperature (T) as a function of time (x): 13 Page 79

14. 14 Page 79

15. h = composition of functions f & g f = outside function, g = inside function 15 Page 79

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

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

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

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

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

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

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

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

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

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

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

27. Inverse Functions 27 Page 80

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

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 2007. (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. 29 Page 80 Example #4

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 30 Page 80

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? 31 Page 80

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? 32 Page 80

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 2007 + 4 = 2011. 33 Page 80

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? 34 Page 80

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) 35 Page 80

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 36 Page 80

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 37 Page 80

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. 38 Page 80

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

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

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

42. Domain & Range of an Inverse Function 42 Page 81

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

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. 44 Page 81

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

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

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

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. 49 Page 81 Example #6

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

53. End of Section 2.4 53