Math gurus, please help!

Discussion in 'Secondary Education' started by Upsadaisy, Apr 18, 2012.

  1. Upsadaisy

    Upsadaisy Moderator

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    Apr 18, 2012

    I'm working with a student in Algebra II. There are a couple of problems in the chapter on log functions that I can't figure out, nor can I find the solutions in the chapter (I sound like a student, myself).

    Here they are:

    Write the inverse function for this


    1. y = log 2x
    2
    The answer given is y = 2^x-1



    2. y = log(x+1)

    No answer given


    Please show me how you get the answers. Thank you.
     
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  3. MikeTeachesMath

    MikeTeachesMath Devotee

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    Apr 18, 2012

    1. I'm assuming the original equation is y = (log base 2 of (x)) + 1.

    f-1(x) is the inverse function.

    log base 2 of x - 1

    Remember: f(f-1(x)) = x

    f(f-1(x)) = log base 2 of (f-1(x)) + 1

    log base 2 of (f-1(x)) = x - 1

    f-1(x) = 2^(x-1)

    ==================

    f(x) = log(x+1)

    log(f-1(x) + 1) = x

    f-1(x) + 1 = 10^x

    f-1(x) = 10^x - 1

    [​IMG]
     
  4. Aliceacc

    Aliceacc Multitudinous

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    Apr 18, 2012

    OK, here's my response:

    First, change each log equation into an exponential equation:
    - keep the base the same.
    - since a log = an exponent, whatever is on the OTHER side of the equal sign from the word "log" is the exponent.
    - the other term is what it equals.

    Second, to find the inverse of a function, switch the x and the y. Solve for y if necessary.

    OK, problem #1.
    To get the answer you list, the problem I'm starting with is:
    y = log (base 2) of (x+1)

    First, switch the problem to exponential form:
    2 to the y = (x+1)
    Now, find the inverse: switch the x and y:
    2 to the (x) = y+1.
    (2 to the x ) -1 = y

    If that wasn't the original equation, let me know.

    Number 2: y = log (base 10)-- that's assumed if there's no other value listed) of (x+1)

    Rewrite in exponential form: 10 to the y = (x+1)
    Now switch the x and y: 10 to the x = (y+1)
    Subtract 1 from both sides to solve for y:

    10 to the x - 1 = y.
     
  5. MikeTeachesMath

    MikeTeachesMath Devotee

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    Apr 18, 2012

    ^ What she said :lol:
     
  6. Upsadaisy

    Upsadaisy Moderator

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    Apr 18, 2012

    What I say is, "Wow", and, "Thanks".

    I understood yours well, Alice. Yes, you both had the equation right. I forgot it wouldn't line up on here.

    Mike, the part I don't understand in your solution is :
    Remember: f(f-1(x)) = x

    I sure didn't remember that. Would you explain it to me (a former middle school math teacher)? Oh, and I really appreciated the photo.

    Thank you both very much.
     
  7. MikeTeachesMath

    MikeTeachesMath Devotee

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    Apr 18, 2012

    It's just a property of the inverse function.

    For example, f(x) = 4x. Therefore f-1(x) = x/4.

    f( f-1(x) ) = f(x/4) = (4x)/4) = x.

    Another example, f(x) = sin(x). Therefore f-1(x) = arcsin(x).

    f( f-1(x) ) = f(arcsin(x)) = sin(arcsin(x)) = x.
     
  8. orangetea

    orangetea Connoisseur

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    Apr 18, 2012

    If you have f(x)=x+2, you know that f-1(x)=x-2

    f(1)=3

    f-1(3)=1

    So, f(f-1(x))=x because f(f-1(3))=f(1)=3
     
  9. orangetea

    orangetea Connoisseur

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    Apr 18, 2012

    Oh, and it goes both ways. If f(x) and g(x) are inverses, then...

    f(g(x)) = x and g(f(x)) = x
     
  10. TeacherGroupie

    TeacherGroupie Moderator

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    Apr 18, 2012

    Since superscripts are impractical on these forums, how about adopting the old convention, from line-printer days, of using a caret to indicate that what follows it is an exponent? Then "x raised to the second power" as x^2, and Mike's f - 1(x) is f^-1(x).
     
  11. Aliceacc

    Aliceacc Multitudinous

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    Apr 18, 2012

    You're welcome, 'Daisy!

    As to the explanation: f( f^-1(x))=x: Finding the inverse means switching the x and y.

    Remember, f^-1(x) means the inverse of x, or y.

    So, to put it into words,
    f( f^-1(x))=x: means take f(the inverse of x)
    or

    Plug the inverse of x into the original function in place of x. As long as f(x) and (f^-1)(x) are inverses, you'll end up with x.

    For what it's worth, this is NOT the explanation I would give to a high school student who is struggling!
     
  12. Upsadaisy

    Upsadaisy Moderator

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    Apr 18, 2012

    Nor to her tutor, who is still struggling.

    My student won't be required to know this, I'm pretty sure. If I didn't think I were helping her, I would bow out, but I know it has been a benefit to her. However, I'm sure not a candidate to teach an algebra II class.
     

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