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Math Help - Finding the equation of a rational function.

  1. #1
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    Finding the equation of a rational function.

    I need help with this problem as soon as possible.

    Suppose you have the function f(x)=2x+b
    cx+d
    This function is undefined at x=-4
    3, the graph of this function strikes the x axis at 8 and the y axis at 1. What are the values for b,c and d?

    I have no idea how to do any of this
    Last edited by mr fantastic; November 9th 2009 at 02:04 AM. Reason: Changed post title
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  2. #2
    Senior Member I-Think's Avatar
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    I'm assuming you mean f(x)=\frac{2x+b}{cx+d}

    Hints
    If the function is undefined at x=\frac{-4}{3}, it usually means a division by zero takes place. *hint*hint

    If it strikes the x axis at 8, that means when f(x)=0, x=8, i.e.:f(8)=0
    And if it strikes the y-axis at 1, that means that at x=0, f(x)=1, i.e. f(0)=1

    Substitute the values and use the hints.
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  3. #3
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    I'm still not fully understanding this...ugh
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  4. #4
    Super Member Bacterius's Avatar
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    Alright, you have :

    f(x)=\frac{2x+b}{cx+d}

    You also know that a division per zero necessarily occurs in this function if x \in R, if and only if c <> 0. You can reasonably assume that c is not equal to zero since that would be really boring. And when a division per zero occurs, what happens ? The function is undefined.

    Therefore, if the function is undefined at x = \frac{ -4}{3}, the denominator is equal to 0, which means you have to solve \frac{ -4}{3} c + d = 0

    The graph of this function cuts the x-axis in x = 8, which means that when f(x) = 0, x = 8. You then have f(8) = 0.

    The graph of this function cuts the y-axis in y = 1, which means that when f(x) = 1, x = 0. You then have f(0) = 1.

    Let's resume all this :

    f(x)=\frac{2x+b}{cx+d}

    \frac{ -4}{3}c + d = 0

    f(8) = 0 , equivalent to \frac{2 \times 8 +b}{c \times 8+d} = 0

    f(0) = 1, equivalent to \frac{b}{d} = 1

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    Now I'll do a bit for you in this spoiler (don't click if you want to find by yourself) :

    Spoiler:
    \frac{b}{d} = 1 means that b = d

    Therefore, the function becomes :

    f(x)=\frac{2x+b}{cx+b}

    Now you know that \frac{ -4}{3} c + d = 0 , which means that c = \frac{ 3d}{4}

    Therefore, the function becomes :

    f(x)=\frac{2x+b}{\frac{ 3d}{4} x+b}

    Remember that b = d, so we have :

    f(x)=\frac{2x+b}{\frac{ 3b}{4} x+b}

    Now use one of the x-axis / y-axis values to substitute, isolate and solve b.

    You can solve for b, therefore you can solve for d, and the last variable remaining ( c) can be found by reverting to the original function and substituting the y-axis cut values.

    I might have done some errors in the spoiler though, so don't just copy blindly.
    Last edited by Bacterius; November 8th 2009 at 08:09 PM.
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  5. #5
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    Quote Originally Posted by thisisnew View Post
    Suppose you have the function f(x)=2x+b
    cx+d
    This function is undefined at x=-4
    3, the graph of this function strikes the x axis at 8 and the y axis at 1. What are the values for b,c and d?

    I have no idea how to do any of this
    when x = -\frac{4}{3}, cx + d = 0, hence
    d - \frac{4c}{3} = 0
    f(x) strikes x-axis at (8,0) and y-axis at (0,1)
    hence 0 = \frac{2(8) + b}{c(8) + d} ,
    hence 16 + b = 0
    b = -16
    also 1 = \frac{2(0) + b}{c(0) + d} ,
    hence c(0) + d = 2(0) + b
    0 + d = 0 + (-16)
    d = -16
    Since d - \frac{4c}{3} = 0 ,
    d = \frac{4c}{3}
    c = (-16)(\frac{3}{4})
    c = -12
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  6. #6
    Super Member Bacterius's Avatar
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    Ukorov, you can simplify a bit your work by clearly saying that b = d, therefore getting rid of the part involving evaluation of d. It makes it a bit shorter, and a lot cooler I find
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