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Thread: Limits #1

  1. #1
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    Limits #1

    Given a function $\displaystyle f$ of a real variable which satisfies the function defined by $\displaystyle f (x) + x \times f(1-x) = 1 +x^2 $, for all real $\displaystyle x$, determine $\displaystyle f(x)$.

    Please explain to me what to do. Ive never done calculus before and this is the question I was given.
    Last edited by mr fantastic; Sep 19th 2009 at 06:16 PM. Reason: OK, now the question is clearer
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  2. #2
    MHF Contributor red_dog's Avatar
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    $\displaystyle f(x)+xf(1-x)=1+x^2$

    Replace $\displaystyle x$ with $\displaystyle 1-x$:

    $\displaystyle f(1-x)+(1-x)f(x)=1+(1-x)^2\Rightarrow f(1-x)=x^2-2x+2-(1-x)f(x)$

    Now replace $\displaystyle f(1-x)$ in the first equality and find $\displaystyle f(x)$.
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