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Thread: Please help!

  1. #1
    Junior Member
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    Smile Please help!

    Consider the function f(x) = Log(1/2)(x^2 - 2x + 3) and determine :


    ( A ) Its domain.
    ( B ) The intervals where it is strictly increasing.
    ( C ) The intervals where it is strictly decreasing.


    Thank you very much for your help!! I appreciate it!
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  2. #2
    MHF Contributor red_dog's Avatar
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    $\displaystyle x^2-2x+3>0, \ \forall x\in\mathbf{R}$.
    So, the domain is $\displaystyle \mathbf{R}$.

    Let $\displaystyle g(x)=\log_{\frac{1}{2}}x$ and $\displaystyle h(x)=x^2-2x+3$.
    Then $\displaystyle f(x)=g(h(x))$
    $\displaystyle g(x)$ is decreasing for all $\displaystyle x>0$.
    $\displaystyle h(x)$ is decreasing for all $\displaystyle x\leq 1$ and increasing for all $\displaystyle x\geq 1$.
    Then $\displaystyle f$ is increasing for all $\displaystyle x\leq 1$ and decreasing for all $\displaystyle x\geq 1$.
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