The range of the function $f\left(x\right)={\log }_{0.5}(x^4-2x^2+3)$ is

$f\left(x\right)={\log }_{0.5}(x^4-2x^2+3)$
$\text{Let}{x}^2=t$
$t^2-2t+3=y$
$\mathrm{D}={\mathrm{b}}^2-4\mathrm{ac}=(-2{)}^2-4\times 1\times 3=-8$
No real roots
y will be always positive
$t_{\min }=-\frac{b}{2a}=\frac{-(-2)}{2\times 1}=1$
$y_{\min }=-\frac{D}{4a}=\frac{-(-8)}{4\times 1}=2$
  Base is less than 1.
Graph will decrease for x > 1.
$\begin{array}{rl} & \therefore \mathrm{f}\left(\mathrm{x}\right)={\log }_{0.}\left(2\right)={\log }_{\left(\frac{1}{2}\right)}\\ & \left(2\right)={\log }_{2^{-1}}=(-1){\log }_{2}2=-1\end{array}$
$\text{Range}\in (-\mathrm{\infty },-1]$