2. RELATION … any set of points

3. RELATION … any set of points { (3,6), (-4,5), (0,2) }

7. Remember … A function is a relationship between variables where each input has exactly one output. Alternate definition = Set of points where each “x” is paired with only one “y”.

8. Is each of these a function? { (2,5), (7,-1), (5,4), (3,8), (-1,4), (0,0) } { (7, 3), (5,-2), (7, 8), (5,9), (0, 4), (6,-3) } { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6) } ( (0,1), (2,1), (-2,1), (-5,1), (7,1) }

9. Is each of these a function? { (2,5), (7,-1), (5,4), (3,8), (-1,4), (0,0) } Yes { (7, 3), (5,-2), (7, 8), (5,9), (0, 4), (6,-3) } No { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6) } No ( (0,1), (2,1), (-2,1), (-5,1), (7,1) } Yes

10. Is each of these a function?

11. Is each of these a function? NoNo YesNo

12. Is each of these a function?

13. Is each of these a function? Yes No Yes No

14. Is each of these a function?

15. You can tell whether a graph is a function using the vertical line test. If a vertical line ever touches the graph more than once, it is NOT a function.

16. If every vertical line just touches the graph once, it IS a function.

17. Is each of these a function?

18. Is each of these a function? NoYesNo No Yes

19. Domain  Set of independent variables in a relation  All the x’s that are used

20. What is the domain? { (1,2), (3,4), (5,6), (7,8), (9,10 } { (3,4), (5,-7), (-1,2), (3, 4), (-1,-1) }

21. What is the domain? { (1,2), (3,4), (5,6), (7,8), (9,10 } { 1, 3, 5, 7, 9 } { (3,4), (5,-7), (-1,2), (3, 4), (-1,-1) } { 3, 5, -1 } {1, 2, 3, 4, 5, 6 } { 3, 7 }

22. Range  Set of dependent variables in a relation  All the y’s that are used

23. What is the range? { (1,2), (3,4), (5,6), (7,8), (9,10 } { (3,4), (5,-7), (-1,2), (3, 4), (-1,-1) }

24. What is the range? { (1,2), (3,4), (5,6), (7,8), (9,10 } { 2, 4, 6, 8, 10 } { (3,4), (5,-7), (-1,2), (3, 4), (-1,-1) } { 4, -7, 2, -1 } { -1, 0, 1, 2, 3, 6 } { 1, 2, 5 }

25. Function Notation Rules or formulas for evaluating functions are often given using notation like … f(x) = 3x + 4 g(x) = x 2

26. We read f(x) as “f of x”. f(x) = 3x + 4 means a function of x is defined with the formula 3x + 4 f(x) basically means “y”.

27. Almost everything in advanced mathematics is defined using function notation.  A function basically gives you a formula that you can plug numbers into to get an answer.

30. Typical problem … f(x) = 3x + 4 Find f(7).

31. Typical problem … f(x) = 3x + 4 Find f(7). Just plug in 7 for x. f(7) = 3  7 + 4 = 21 + 4 = 25

32. f(x) = 5x – 1 g(x) = 2x Find  f(3)  g(4)  f(-2)  g(-8)  f(0)

33. f(x) = 5x – 1 g(x) = 2x Find  f(3)14  g(4)8  f(-2)-11  g(-8)-16  f(0)-1

34. k(x) = x 2 + 3x – 5 Find k(4)

35. k(x) = x 2 + 3x – 5 Find k(4) 4 2 + 3  4 – 5 16 + 12 – 5 = 23

36. p(t) = 180 – 16t 2 Find p(3)

37. p(t) = 180 – 16t 2 Find p(3) 180 – 16  9 = 36

38. m(x) = 4x + 3 Domain = { 1, 2, 3, 4, 5 } Find the range.

39. m(x) = 4x + 3 Domain = { 1, 2, 3, 4, 5 } Find the range. Range = { 7, 11, 15, 19, 23 }

40. g(x) = { (2,5), (-1,4), (6,0) } Find g(2) g(6) g(-1)

41. g(x) = { (2,5), (-1,4), (6,0) } Find g(2) = 5 g(6) = 0 g(-1) = 4

42. f(x) = Find f(-2), f(-1), f(0), f(2)

44. f(x) = 5x – 1 g(x) = 2x Find  f[g(3)]  g[f(9)]

45. f(x) = 5x – 1 g(x) = 2x Find  f[g(3)] = f[6] = 29  g[f(9)] = g[44] = 88

46. g(x) = 5x – 3 If g(x) = 62, what is x? If g(x) = -98, what is x?

47. g(x) = 5x – 3 If g(x) = 62, what is x? 5x – 3 = 62 x = 13 If g(x) = -98, what is x? 5x – 3 = -98 x = -19