A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

  • This Question is Closed
  1. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  2. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  3. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  4. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    plug in 3 to every (x)

  5. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  6. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  7. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    You have it. Let's see it.

  8. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\frac{ -3x - 7/ x - 1 - 7}{ -3x - 7/ x - 1 +3 } \]

  9. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  11. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  12. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    for the numerator do i distribute?

  14. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  16. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Okay, now tackle the denominator.

  17. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  18. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    be prepared to do a raft of algebra ready?

  19. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    very ready!

  20. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    first we compute \[f(g(x)) = x\]

  21. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  22. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  24. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  25. tkhunny
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    THERE'S that denominator!

  26. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  27. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    you need the first step?

  29. SolomonZelman
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  30. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  31. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok lets go slow

  32. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  33. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  34. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    then \[\frac{ -3x - 7 - 7x + 1 }{ -3x - 7 + 3x - 1 }\]

  35. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    forgot that distributive law already huh?

  36. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    distribute the \(-7\) up top and the \(3\) below

  37. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  38. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  39. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh no

  40. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    what can you get with \[-7(x-1)\]when you distribute?

  41. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  42. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yay

  45. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  47. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    let me try to write it

  48. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  49. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  50. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    should have used parentheses around the first term in the top

  51. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    multiply by \(x+3\) top and bottom what do you get before distributing etc

  52. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    this might be trickier because of the \(-3\) up top so be careful with parentheses

  53. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  54. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes

  55. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now get rid of the parentheses

  56. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    -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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  59. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.