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.

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2. $x^2-2x+3>0, \ \forall x\in\mathbf{R}$.
So, the domain is $\mathbf{R}$.

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