## anonymous one year ago Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

1. tkhunny

"Confirm" -- This is nice. Just follow the instructions. That's what "confirm" is all about.

2. anonymous

$f(x) = \frac{ x-7 }{ x+3 } ; g(x)=\frac{ -3x-7 }{ x-1 }$

3. tkhunny

Well? If it were f(3), what would you do?

4. anonymous

plug in 3 to every (x)

5. tkhunny

Do that with g(x) and you will have f(g(x)). It's just what the notation means.

6. anonymous

so for every (x) in f(x) i plug in -3x - 7 / x - 1 ?

7. tkhunny

You have it. Let's see it.

8. anonymous

$\frac{ -3x - 7/ x - 1 - 7}{ -3x - 7/ x - 1 +3 }$

9. tkhunny

Horrid notation. Please use parentheses to clarify intent. f(x) = (x-7)/(x+3) g(x) = (-3x-7)/(x-1) f(g(x)) = ([(-3x-7)/(x-1)]-7)/([(-3x-7)/(x-1)]+3) It's not pretty, but it's complete and accurate. Now, for your best algebra skills.

10. anonymous

my apologies. can i cross out ( -3x - 7 / x - 1 ) ?

11. anonymous

or do i multiply ( x - 1 ) for numerator and denominator?

12. tkhunny

You simplify. Let's take the numerator. $$\dfrac{-3x-7}{x-1} - 7 = \dfrac{-3x-7}{x-1} - \dfrac{7(x-1)}{x-1} = \dfrac{(-3x-7)-7(x-1)}{x-1}$$ Keep going. We're just adding fractions.

13. anonymous

for the numerator do i distribute?

14. tkhunny

You do what it takes to simplify it. If the Distributive Property is appropriate, then do that. $$= \dfrac{-3x - 7 - 7x + 1}{x-1}$$ One step at a time.

15. anonymous

from that i got $\frac{ -10x - 6 }{ x - 1 }$

16. tkhunny

Okay, now tackle the denominator.

17. anonymous

could you start me off so i know where to begin?

18. anonymous

be prepared to do a raft of algebra ready?

19. anonymous

20. anonymous

first we compute $f(g(x)) = x$

21. anonymous

$f(g(x))=f(\frac{ -3x-7 }{ x-1 })$

22. anonymous

now we are going to replace all $$x$$ in $$f(x)$$ by $$\frac{ -3x-7 }{ x-1 }$$ that is actually very easy for me to do here, by cutting and pasting

23. tkhunny

You already have the denominator. You MUST show some algebra. ([(-3x-7)/(x-1)]+3) = $$\dfrac{-3x-7}{x-1}+3$$ Go!

24. anonymous

$f(x) = \frac{ x-7 }{ x+3 }$so $f(g(x))= \frac{ \frac{ -3x-7 }{ x-1 }-7 }{ \frac{ -3x-7 }{ x-1 }+3 }$

25. tkhunny

THERE'S that denominator!

26. anonymous

to get rid of that annoying compound fraction, multiply the numerator and denominator by $$x-1$$ (carefully using parentheses)

27. anonymous

then multiply out, combine like terms since you know the answer will just be $$x$$ you should expect an orgy of cancellation at the last couple steps

28. anonymous

you need the first step?

29. SolomonZelman

(I would rather just find the inverse, without doing f(g(x))=x and vv, but ... $$-:($$ )

30. anonymous

uh, i got $\frac{ -10x - 6 }{ 8 }$

31. anonymous

ok lets go slow

32. anonymous

$\frac{ \frac{ -3x-7 }{ x-1 }-7 }{ \frac{ -3x-7 }{ x-1 }+3 }$ the $$x-1$$ will cancel top and bottom to get $\frac{-3x-7-7(x-1)}{-3x-7+3(x-1)}$

33. anonymous

now multiply out using the almighty distributive law then combine like terms

34. anonymous

then $\frac{ -3x - 7 - 7x + 1 }{ -3x - 7 + 3x - 1 }$

35. anonymous

forgot that distributive law already huh?

36. anonymous

distribute the $$-7$$ up top and the $$3$$ below

37. anonymous

starting here $\frac{-3x-7-7(x-1)}{-3x-7+3(x-1)}$

38. anonymous

(-3x - 7 ) ( -7x + 1) (-3x - 7 ) ( 3x - 1)

39. anonymous

oh no

40. anonymous

what can you get with $-7(x-1)$when you distribute?

41. anonymous

-7x + 7 OMG -.- im so disappointed with myself right now.

42. anonymous

ok so lets back up to $\frac{-3x-7-7(x-1)}{-3x-7+3(x-1)}$ and see what we get when we remove the parentheses by "we" i mean "you"

43. anonymous

i got, -10x / - 10 which would equal to (x) so we have f(g(x)) out of the way, i started g(f(x)), coud you guide me with that to please?

44. anonymous

yay

45. anonymous

ok sure lets start just as before with an annoying compound fraction but try it yourself first, it is going to work almost exactly like this one

46. anonymous

$(-3 (\frac{ x-7 }{ x+3 }) - 7 (numerator) ; (\frac{ x-7 }{ x+3}) - 1 (denominator)$

47. anonymous

let me try to write it

48. anonymous

yeah actually that looks good now multiply top and bottom but this time by $$x+3$$ instead of $$x-1$$

49. anonymous

$g(x)=\frac{ -3x-7 }{ x-1 }$ $g(f(x))=\frac{ -3 \frac{ x-7 }{ x+3 } -7 }{ \frac{ x-7 }{ x+3 } -1 }$

50. anonymous

should have used parentheses around the first term in the top

51. anonymous

multiply by $$x+3$$ top and bottom what do you get before distributing etc

52. anonymous

this might be trickier because of the $$-3$$ up top so be careful with parentheses

53. anonymous

-3(x-7) - 7(x+3) / (x-7) - 1(x+3) ?

54. anonymous

yes

55. anonymous

now get rid of the parentheses

56. anonymous

you should of course just be left with some x up top and a number in the bottom that will cancel with it

57. anonymous

-10x / - 10 like before we get (x) for g(f(x)). whew! long problem lol. but thank you sooooo much for all of your time & help!! i wish i could give you more than just a medal & a fan!

58. anonymous

glad to help, hope you learned something (at least learned how to do these!) who studies math in late august?