Range of the function ${log}_{0.5} (x^{4} - 2 x^{2} + 3)$ is

(- \infty, 0)

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(- \infty, 1)

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(- \infty, - 1]

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(- \infty, \frac{1}{2}]

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Solution

The correct option is C $(- \infty, - 1]$
$x^{4} - 2 x^{2} + 3$
$= x^{4} - 2 x^{2} + 1 + 2$
$= (x^{2} - 1)^{2} + 2$
Clearly, $D (f)$ is $R$

Now, $2 \leq (x^{2} - 1)^{2} + 2 < \infty$
As ${log}_{a} x$ is strictly decreasing for $0 < a < 1$ ,
$- \infty < {log}_{1 / 2} (x^{4} - 2 x^{2} + 3) \leq {log}_{1 / 2} 2$
$\Rightarrow - \infty < {log}_{1 / 2} (x^{4} - 2 x^{2} + 3) \leq - 1$

Hence, $R (f) = (- \infty, - 1]$

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