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Thread: Composition & Functoins

  1. #1
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    Composition & Functoins

    hey all, need help with a problem

    Question: Use composition to show that f(x) = (x + 3) / (x - 2) and g(x) = (2x - 3) / (x - 1) are inverse of each other.

    Any help would be greatly appreciated, thanks

    -Laconian
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  2. #2
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    What means that f and g are inverse of each other?
    Just that $\displaystyle \forall x \in Dom(g), f(g(x))=x$ and $\displaystyle \forall x \in Dom(f), g(f(x))=x$

    For example, let $\displaystyle x \in \mathbb{R}-\{1\},$

    $\displaystyle f(g(x))$
    $\displaystyle =f(\frac{2x-3}{x-1})$
    $\displaystyle =\frac{\frac{2x-3}{x-1}+3}{\frac{2x-3}{x-1}-2}$
    $\displaystyle =\frac{2x-3+3(x-1)}{2x-3-2(x-1)}$
    $\displaystyle =\frac{5x-6}{-1}$ !!! that's not what we wanted!

    But if we change $\displaystyle f(x)=\frac{x+3}{x-1}$ by $\displaystyle f(x)=\frac{x-3}{x-1}$, as you can see, that would work. Then try with $\displaystyle gof$.
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