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Math Help - Finding the function that will satisfy the equation.

  1. #1
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    Finding the function that will satisfy the equation.

    If g(x) satisfies 4g(x) + g(1-x) = 3x2, what is what is g(x)?

    I'm really stuck, what is a possible solution?
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  2. #2
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    Re: Finding the function that will satisfy the equation.

    I expect since this particular combination of functions gives you a quadratic, you will probably find that g(x) is a quadratic. So let \displaystyle \begin{align*} g(x) = a\,x^2 + b\,x + c \end{align*}, then \displaystyle \begin{align*} g(1 - x) = a(1 - x)^2 + b(1 - x) + c = a - 2a\,x + a\,x^2 + b - b\,x + c \end{align*}. So

    \displaystyle \begin{align*} 4g(x) + g(1 - x) &= 3x^2 \\ 4a\,x^2 + 4b\,x + 4c + a - 2a\,x + a\,x^2 + b - b\,x + c &= 3x^2 \\ 5a\,x^2 + (3b - 2a)x + a + b + 5c &= 3x^2 + 0x + 0 \end{align*}

    So this means \displaystyle \begin{align*} 5a = 3 \implies a = \frac{3}{5}, 3b - 2a = 0 \implies b = \frac{2}{5}, a + b + 5c &= 0 \implies c = -\frac{1}{5} \end{align*}.

    Therefore a function which satisfies your original equation is \displaystyle \begin{align*} g(x) = \frac{3}{5}x^2 + \frac{2}{5}x - \frac{1}{5} \end{align*}.
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  3. #3
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    Lightbulb Re: Finding the function that will satisfy the equation.

    Simple solution:

    Substitute x\to 1-x...you get a second equation, eliminate g(1-x) between them to obtain g(x):

    g(x)\text{:=}\frac{1}{15} \left(9 x^2+2 x-3\right)
    Last edited by MaxJasper; August 24th 2012 at 07:00 PM.
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  4. #4
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    Re: Finding the function that will satisfy the equation.

    Replace x with 1-x to obtain

    4g(1-x) + g(x) = 3(1-x)^2. We already know

    4g(x) + g(1-x) = 3x^2

    Multiply the second equation by -4 and add them.

    -15g(x) = 3(1-x)^2 - 12x^2

    Divide both sides by -15 and simplify.
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