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Thread: Please help with this Pre-calculus question

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    Please help with this Pre-calculus question

    If f(3)=M+1 for some rational/reciprocal function f(x) with the denominator of (x+2). If g(x) is the inverse function of f(x) and g(m)=2, find a possible function f(x) in the form f(x)= (ax+b)/(cx+d). There are many possible answers.
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    Re: Please help with this Pre-calculus question

    Quote Originally Posted by FrancisG View Post
    If f(3)=M+1 for some rational/reciprocal function f(x) with the denominator of (x+2). If g(x) is the inverse function of f(x) and g(m)=2, find a possible function f(x) in the form f(x)= (ax+b)/(cx+d). There are many possible answers.
    You have posted the same question twice.
    That is strictly forbidden on this helpsite.
    Please follow the rules in future postings.
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    Re: Please help with this Pre-calculus question

    Sorry I posted this to the wrong forum so I re-posted it and forgot to delete this one.
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    Re: Please help with this Pre-calculus question

    are $M$ and $m$ intended to be distinct numbers, or does $M=m$ ?
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    Re: Please help with this Pre-calculus question

    Quote Originally Posted by romsek View Post
    are $M$ and $m$ intended to be distinct numbers, or does $M=m$ ?
    suppose $M$ and $m$ are abitrary

    $f(3)=M+1$

    $f(x) = \dfrac{a x + b}{x+2}$

    $g(x) = f^{-1}(x)$

    $g(m)=2$

    ok, let's digest all of this

    $f(3) = M+1 = \dfrac{3a+b}{5}$

    $5M+5 = 3a+b$

    $g(m)=2 \Rightarrow f(2)=m$

    $f(2)=m \Rightarrow \dfrac{2a+b}{4}=m$

    $2a+b = 4m$

    so now we have two equations in $a,~b$

    $\begin{pmatrix}3 &1\\2 &1\end{pmatrix}\begin{pmatrix}a \\ b\end{pmatrix} = \begin{pmatrix}5(M+1)\\4m\end{pmatrix}$

    and this can be solved to produce

    $a=5-4m+5M,~b=-10+12m-10M$

    this results in

    $f(x) = \dfrac{(5-4m+5M)x + (-10+12m-10M)}{x+2}$
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