## Inopeki 3 years ago TuringTest, functions?

1. TuringTest

Right, so now that we've covered a lot of that let's look for something tricky: starting simple: y=x+7 is y a function of x?

2. Inopeki

Yes

3. TuringTest

good now, is x a function of y ?

4. Inopeki

No

5. Inopeki

Cause x is the dependent variable

6. TuringTest

x is the independent variable, but the independent variable can be a function of the dependent variable sometimes. It can work both ways. Not always, though. You must check. to find out try solving for x above

7. Inopeki

y=x+7 y-7=x?

8. TuringTest

so is x a function of y?

9. TuringTest

can we switch their places?

10. Inopeki

They should work both ways.

11. TuringTest

right, so in y=x+7 we can write one variable as a function of the other at will x=y-7 so we can change the way we look at this and call it f(x)=x+7=y or g(y)=y-7=x they are both functions and both equivalent. now trickier: what about y=x^2 is y a function of x ?

12. Inopeki

Yes?

13. TuringTest

yes, each x gives exactly one y now the hard part :P is x a function of y?

14. Inopeki

No, cause each y gives us 2 x

15. Inopeki

It could be a function of yx

16. TuringTest

how so? elaborate ;)

17. zbay

|dw:1326051567892:dw| every element of a is maped to an element of b

18. Inopeki

y=x^2 yx=x Therefore each y gives us 2 x It is a function of yx

19. TuringTest

nice try but not, the algebra is wrong btw zbay's drawing show essentially the same concept as the tables I made y=x^2 solve for x do you know how?

20. Inopeki

y/x=x actually

21. Inopeki

Lol i rushed it

22. TuringTest

that's not solved for x, you have x on both sides :/

23. TuringTest

take the square root of both sides

24. Inopeki

y1/2=x?

25. TuringTest

is the 1/2 and exponent?

26. TuringTest

an exponent*

27. Inopeki

Yes, didnt you say that yesterday?

28. TuringTest

yes, just making sure, you should write it as y^(1/2)=x to avoid confusion that is right but you forgot one little detail about taking a square root of both sides of an equation...

29. Inopeki

ALright

30. TuringTest

plus or minus...

31. Inopeki

What?

32. TuringTest

when you solve$x^2=4$there are two answers, remember? one positive and one negative. or did you not know that?

33. Inopeki

No, why?

34. pokemon23

TURNING TEST BUDDY :D

35. TuringTest

look at two different ways of getting the answer:$x^2=y$look at y=4:$2^2=4$$(-2)^2=4$so x=2 and x=-2 BOTH correspond to y=4 This is fine for y as a function of x, but means that x cannot be a function of y, because one y corresponds to multiple x.

36. TuringTest

hi pokemon :)

37. Inopeki

ooooh

38. Inopeki

since negative multiplied by negative is positive

39. TuringTest

exactly :D

40. TuringTest

so whenever you have to take the square root like that we must write$x^2=y\to x=\pm\sqrt y$which shows that x is not a function of y sometimes we can ignore this and only look at the positive root to make it a function, but we may have to adjust things accordingly to do that. Here is something many tutors on Open Study still don't know so listen up...

41. TuringTest

If you are asked What is $\sqrt4$there is only one answer, 2 but if you are are asked $x^2=4$what is x? you must look at both plus and minus:$x^2=4\to x=\pm2$ Many tutors here think that you always need +/- when you look at a square root. The truth is you only do that when you have to TAKE the square root of both sides of an equation. There, be a step ahead ;)

42. TuringTest

so that said, back on topic$r^2=t$is t a function of r? is r a function of t?

43. Inopeki

r is a function of t but not vice versa

44. TuringTest

actually vice versa, but not vice versa lol

45. TuringTest

which is a function of what? careful.

46. Inopeki

So t is a function of r?

47. TuringTest

yes, because it's okay for t to be the dependent variable. one r gives one t If we turn it around and solve for r however, we have to use +/-sqrt so that means r won't be a function of t.$r^2=t\to r=\pm\sqrt t$so each value of t except zero will give us two real answers. not a function

48. Inopeki

Ohh

49. TuringTest

each positive value of t will give two real answers*

50. Inopeki

Yeah, throw me another one

51. TuringTest

what about$r=\sqrt t$???

52. Inopeki

sqrt(t) cant be the dependent variable

53. TuringTest

actually it can because I took away the +/- sign remember what I said about how many tutors are wrong about square roots? in the case of$\sqrt4=2$we have one answer, but for$x^2=4\to x=\pm2$we have two answers. so this is equivalent to the first case, we are only looking at the /positive/ square root.

54. Inopeki

Oh, so if the +/- sign is gone then it can me the other way around?

55. TuringTest

it is the act of TAKING the square root that introduces the +/-, which is what makes it not a function. If we don't have to actually take the square root, then it is just positive. so$r=\sqrt t$is a function

56. TuringTest

yes because we are only looking at the positive root I'll draw...

57. TuringTest

I made our graph with r as the vertical direction so we could use the vertical line test. Think of how that works.|dw:1326053937990:dw|I'll draw in r=sqrt(t)...

58. TuringTest

Actually first the graph of r^2=t:|dw:1326054064202:dw|

59. TuringTest

Now the graph of just$r=\sqrt t$|dw:1326054150478:dw|see how we just use the positive part? notice how r^2=t does not pass the vertical line test with t as the independent variable, so r is not a function of t however in our last graph we clearly do have r=sqrt(t) with r as a function ot t.

60. TuringTest

|dw:1326054308550:dw|^not a function|dw:1326054323340:dw|^is a function

61. Inopeki

because of the plus/minus it can be |dw:1326054372813:dw| or |dw:1326054351716:dw| Depending on if its negative or positive

62. TuringTest

but that would be splitting the graph up from it's original form if we start Let me try to convey the idea graphically and with the table...

63. Inopeki

So that was wrong?

64. TuringTest

Nothong really I just want to clear up the idea of what it means to break up the function like that. say we have$y=x^2$|dw:1326054620838:dw|you would agree that y is a function of x, yes?

65. Inopeki

No its not

66. TuringTest

because it passes the vertical line test|dw:1326054781944:dw|but what if we changed the coordinates on the graph put x vertical and y horizontal|dw:1326054851735:dw|even though the function hasn't changed, by changing the axis and using the vertical line test we show that y is a function of x, but not vice versa

67. Inopeki

ut its x, not y. Shouldnt it be horizontal line test?

68. TuringTest

no, vertical line test for a graph of independent variable on the horizontal dependent on the vertical however doing a horizontal line test would amount to the same thing as changing the axes if you noticed ;)|dw:1326055073462:dw|same result, we see that y is a function of x, but not the other way around.

69. TuringTest

so another way to see it, is|dw:1326055195330:dw|is y a function of x?

70. Inopeki

Yes

71. TuringTest

good is x a function of y?

72. Inopeki

no

73. TuringTest

74. Inopeki

|dw:1326055271855:dw|

75. TuringTest

perfect :)

76. Inopeki

:D

77. TuringTest

last thing is easier, tables... x | y ---- 2 | 2 5 | 1 0 | 3 4 | 6 is y a function of x ? is x a function of y ?

78. Inopeki

y is a function of x and x is a function of y

79. TuringTest

excellent actually I should be more careful with my language. With the table we can only say that this is potentially a function because we don't necessarily have all the values. That's a technicality though. ok last one...

80. TuringTest

r | t --- 0 | 1 2 | 3 5 | 9 7 | 3 4 | -14 is r(t) a function? is t(r) a function?

81. Inopeki

No Yes

82. TuringTest

correct! very nice, I think you understand the concept of functions :D

83. Inopeki

Seriously? :D

84. TuringTest

yes, you are right :D care to state your rationale?

85. Inopeki

Rationale?

86. TuringTest

87. TuringTest

how did you know?

88. Inopeki

89. TuringTest

yeah r | t --- 0 | 1 2 | 3 5 | 9 7 | 3 4 | -14 is r(t) a function? is t(r) a function WHY?

90. Inopeki

r | t --- 0 | 1 2 | 3<-\ 5 | 9 These make t have several values of the same variable (2,7) I dont see 7 | 3<-/ duplicates on the other side. 4 | -14

91. TuringTest

right, so r(t) does not represent a function, where t(r) does perfect!!!!

92. Inopeki

Yeah!!! :DDD

93. TuringTest

so now you have three interpretations of a function: graphical, numerical, and mathematical. try to harmonize them in your mind, they will be your best friends :)

94. Inopeki

Ok :D Im going to take a 30min break now :)

95. TuringTest

well deserved, take your time :D