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Thread: 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; Nov 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 $\displaystyle f(x)=\frac{2x+b}{cx+d}$

    Hints
    If the function is undefined at $\displaystyle x=\frac{-4}{3}$, 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 :

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

    You also know that a division per zero necessarily occurs in this function if $\displaystyle x \in R$, if and only if $\displaystyle c <> 0$. You can reasonably assume that $\displaystyle c$ 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 $\displaystyle x = \frac{ -4}{3}$, the denominator is equal to $\displaystyle 0$, which means you have to solve $\displaystyle \frac{ -4}{3} c + d = 0$

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

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

    Let's resume all this :

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

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

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

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

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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:
    $\displaystyle \frac{b}{d} = 1$ means that $\displaystyle b = d$

    Therefore, the function becomes :

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

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

    Therefore, the function becomes :

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

    Remember that $\displaystyle b = d$, so we have :

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

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

    You can solve for $\displaystyle b$, therefore you can solve for $\displaystyle d$, and the last variable remaining ($\displaystyle c$) 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; Nov 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 $\displaystyle x = -\frac{4}{3}$, cx + d = 0, hence
    $\displaystyle d - \frac{4c}{3} = 0$
    f(x) strikes x-axis at (8,0) and y-axis at (0,1)
    hence $\displaystyle 0 = \frac{2(8) + b}{c(8) + d}$ ,
    hence 16 + b = 0
    b = -16
    also $\displaystyle 1 = \frac{2(0) + b}{c(0) + d}$ ,
    hence c(0) + d = 2(0) + b
    0 + d = 0 + (-16)
    d = -16
    Since $\displaystyle d - \frac{4c}{3} = 0$ ,
    $\displaystyle d = \frac{4c}{3}$
    $\displaystyle c = (-16)(\frac{3}{4})$
    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 $\displaystyle b = d$, therefore getting rid of the part involving evaluation of $\displaystyle d$. It makes it a bit shorter, and a lot cooler I find
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