## srw1496 2 years ago Why isn't the inverse of a function always a function?

1. ghazi

okay just suppose that you have a function lets say y= sin x now inverse of it will be $x= \sin^{-1}y$ so does the definition of function applies in inverse, try it ?

2. ghazi

did you get it?

3. srw1496

uhm no,

4. ghazi

for defining a function it should have one output to the one input but when you will reverse the condition that is inverse , you will have same x for two Ys which contradicts the definition of function

5. ghazi

and in some cases it is possible the inverse of a function could be a function like if i say y=f(x)=x its inverse could be a function but if i say y= x^3 its a function but inverse is not a function

6. ghazi

still confused? @srw1496

7. srw1496

uh yeah...

8. satellite73

i am confused if $$f(x)=x^3$$ then $$f^{-1}(x)=\sqrt[3]{x}$$ which is a function

9. ghazi

@satellite73 i was missing you and i knew this

10. ghazi

@srw1496 look fundamentally just check out the definition of function and if the inverse of function gives two values for same input then it is not a function

11. ghazi

and yes my apology for choosing wrong example

12. satellite73

on the other hand, lets look at $$f(x)=x^2$$ here we have $$f(2)=2^2=4$$ and $$f(-2)=(-2)^2=4$$ as well so both $$(-2,4)$$ and $$2,4)$$ are on the graph but if you switch the ordered pairs, then you get both $$(4,-2)$$ and $$(4,2)$$ so the inverse is not a function

13. satellite73

the difference is $$x^3$$ is a "one to one" function, which means two different inputs give two different outputs but $$x^2$$ is not a one to one function, different inputs can give the same output

14. ghazi

y=x^2 is not a function

15. Kainui

q is not a letter

16. satellite73

????!!!!

17. Kainui

What are you talking about, seriously?

18. Kainui

First you tell me arcsine isn't a function and now this? lol

19. satellite73

this is not a pipe

20. ghazi

yes plot y=x^2 and use straight line test , straight line cuts the parabola twice so its not a function

21. ghazi

lets play

22. Kainui

|dw:1356208624058:dw|

23. malevolence19

y=x^2 is a function. x=sqrt(y) is not for all y. Only positive y.

24. ghazi

25. ghazi

LOL

26. srw1496

uhhh i have no idea what is going on...

27. malevolence19

It's not injective. That doesn't mean it isn't a function...

28. ghazi

we are having useless debate but @srw1496 i think @satellite73 has explained everything that is all you need to know

29. Kainui

You cut yours the wrong way, kid. Study my picture carefully and you'll see.

30. ghazi

31. satellite73

@srw1496 of course you are confused, this was a silly discussion post again, ignore all this

32. ghazi

here you go , the first one , or else you can google it @Kainui

33. malevolence19

Its as simple as this. For a function you put in an x, you get out a y. The say X will not give you more than 1 Y. To test for it being "one to one" we do the "horizontal line test". So if we look at y=x^2 we get:|dw:1356208787580:dw| Since we know an inverse is just rotating it around the line y=x we have the inverse **would** look like this: |dw:1356208824688:dw| But the vertical line test (how we test if its a function) FAILS! To get around this we usually just define it over a certain domain. So,y=sqrt(x) for positive x.

34. ghazi

@Kainui this is the another one http://www.sparknotes.com/math/algebra1/quadratics/section1.rhtml , so before having any debate , do know what we are talking about

35. malevolence19

for positive y***.

36. dpaInc

lol...

37. Kainui

You said y=x^2 is not a function lol

38. ghazi

yes i said that and i proved it

39. Kainui

No you proved x=y^2 is not a function

40. ghazi

no no no dont change the point please, go through the comments and the links i have mentioned , its not a function that is y=x^2 and i am done , thank you :)

41. Kainui

Lol alright but you're backwards man

42. malevolence19

ghazi you're a fluttering idiot. Get out of here. y=x^2 is a function, stop trying to be a fluttering troll.

43. Kainui

I love how it censors the fluttering words with owl pellet

44. ghazi

@malevolence19 look dude i am not violent yet and dont provoke me to become violent be like a good kid and again for your satisfaction here is your answer http://answers.yahoo.com/question/index?qid=20071023192855AAs2E9z mind your tongue guys

45. malevolence19

AGAIN. THAT IS X=Y^2 NOT Y=X^2 dumbflutter

46. ghazi

thanks, i am not accepting, do accept what you want

47. malevolence19

I get a warning? Really, get this guy out of here who is flat out spreading BS.

48. malevolence19

At least my response is right.

49. ghazi

just plot y=x^2 and see the result , secondly you are not supposed to use bad words

50. malevolence19

I have plotted it, hundreds of times, YOU plot it. http://www.wolframalpha.com/input/?i=y%3Dx^2

51. AccessDenied

My very first note of the thread is seeing that language I have not even read the whole question yet.

52. ghazi

:P

53. AccessDenied

It would be correct, however, to say y = x^2 is a function because of the vertical line test. Its inverse, y = + or - sqrt(x), is not a function because it fails the vertical line test. The lines must be vertical because we are only concerned that the function does not output two y-values for any single x-value. Horizontal lines only decide if the inverse is a function.

54. Kainui

Yeah I would have to agree with @malevolence19 that ghazi is intentionally misleading.

55. ghazi

alright guys its over, why are you dragging it , it doesnt sound fun to me at all

56. ParthKohli

As I said in the other one, let$\{(1,2),(2,2),(3,2)\cdots\}$be a function. Then the inverse is$\{(2,1),(2,2),(2,3)\cdots\}$which overrules the definition of a function