3. Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function of degrees °F Functions: What the f?

4. Function notation y = 1 - x 2 f(x) = 1 - x 2 “y is a function of x” function notation f(x) is read “f of x”

5. Evaluating functions f(x) = 1 - x 2 What is f(-1)? In other words, evaluate the function f(x) at x = -1. f(-1) = 1 – (-1) 2 = 1 – 1 = 0

7. Domain: For what values of x is the function defined? Range: For what values of y is the function defined?

8. Basic examples f(x)=ax 2 +bx+c {all real numbers} g(x)=1/x {x: x  0} *denominator cannot be 0 h(x)=√x {x: x ≥ 0} v(x)=ln(x) {x: x > 0}

9. **** Notation time-out****** In mathematics, : means “such that” Ex: {x : x  0} means “the set of all x’s such that x  0”

10. Domain of y =  2x + 3 {x: 2x + 3  0} “all x’s such that 2x+3 is greater than or equal to 0”

11. Domain of y = {x: 2x + 3  0} Or {x: x  -3/2} 1 2x + 3

12. Domain of y =  2x + 3 All real numbers 3

13. Domain of y = ln(2x + 3) { x: 2x + 3 > 0 }

14. Domain of y = { x: 2x + 3 > 0 } 1  2x + 3

15. Assignment A p.27/ 1,2,(3-23)odd,37

16. SAT Problem of the Day! “700 on the SAT math!!! Heck yes, I am going to UCLA!”

17. * A more CONCISE way to describe sets. ** Is used interchangeably with set notation to express domain and range.

18. {x:…}INTERVAL x  3 x  3 x  3 x  3 x  3 (- , 3) (- , 3] (3,  ) [3,  ) (- , 3) & (3,  )

19. -2  x  3 -2  x  3 -2  x  3 -2  x  3 All reals (- ,  ) (- 2, 3] [- 2, 3) (- 2, 3) [- 2, 3]

20. MIX AND MATCH!!!

21. Assignment B Ignore the directions. Instead… Find the domain of each function and write it using a) set notation and b) interval notation. p.27-28/ 4,6,8,20,30,34

22. SAT Problem of the Day!

24. y = |x| Absolute value

25. y = x 2 parabola

26. y =  x Even root

27. y =  x 3 Odd root

28. 1 y = e x Exponential growth

29. 1 y = e - x Exponential decay

30. 1 y = ln x

31. y = 1/x

32. y = 1/x 2

33. y = x 2 - 2 y = x 2 y = x 2 + 3 + move up - move down

34. y = |x| y = |x - 6| y = |x +3| + move left - move right

35. y = log 2 x y = log 2 (x + 2) left 2

36. y =  x y = -  x Flip about x

37. 1 y = e x y = - e x flip

38. 1 y = - ln x

39. 1 y = - e -x

40. Given the graph of y = f(x), To graph y = f(x) ± a, Move the graph of y = f(x) up/down a units

41. Given the graph of y = f(x), To graph y = f(x ± a), Move the graph of y = f(x) left/rt a units + is left, - is right!

42. Given the graph of y = f(x), To graph y = -f(x), flip the graph of y = f(x) with respect to the x-axis.

43. The graph of y = - f(x) is flipped about the x-axis. The graph of y = f(-x) is flipped about the y-axis.

44. y =  -x

45. 1 y = e - x

46. 1 y = ln (-x)

47. Given the graph of y = f(x), To graph y = kf(x), Multiply all the y values of y = f(x) by k. Steeper if k > 1. Flatter if k < 1

48. y =  x y = 2  x steeper

49. y = |x| y = 2|x| y = ½ |x|

50. y =  x 3 y = 2  x 3 double the y-values

51. y =  x 3 y = ½  x 3 half the y-values

52. 1 y = e x y = 2 e x 2

53. Assignment C p. 28/ 47-55,83,93

54. SAT Problem of the Day!

56. y = |x| y = -|x - 4| + 3 flip right 4 up 3 y = -|x - 4| + 3

57. y = |x| y = -2|x - 4| + 3 flip right 4 up 3 y = -2|x - 4| + 3 steeper

58. y = x 2 y = -(x - 6) 2 + 1 flip right 6 up 1 y = -(x - 6) 2 + 1

59. SAT Problem of the Day

61. Given two functions y=f(x) and y=g(x). “f composed with g of x”

62. Example: Finding composite functions. Given f(x)=2x+3 and g(x)=cos(x). Find…

63. Example: Finding composite functions. Given f(x)=2x+3 and g(x)=cos(x).

64. Try this… Find two functions f and g such that F(x)=

66. Even vs. Odd ODD functions are symmetric with respect to the origin. EVEN functions are symmetric with respect to the y- axis.

67. Even or Odd?

73. Even and Odd Functions Even, odd, or neither test: The function y=f(x) is EVEN if f(-x) = f(x). The function y=f(x) is ODD if f(-x) = -f(x). Otherwise, it is neither even nor odd.

74. Example Determine whether the function is even, odd, or neither. a) b) odd * cos is even even

75. Assignment E p. 28-29/ 57 – 70 *Test on Unit P: Functions This Thursday! Review Assignment online www.geocities.com/mskadlac