The range of the function f(x) = log_{0.5}(x^ | vedclass

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(B) Let $g(x) = x^4 - 2x^2 + 3 = (x^2 - 1)^2 + 2$. Since $(x^2 - 1)^2 \geq 0$, the minimum value of $g(x)$ is $2$ (at $x^2 = 1$) and the maximum value

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$(-\infty, -1]$

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(B) Let $g(x) = x^4 - 2x^2 + 3 = (x^2 - 1)^2 + 2$. Since $(x^2 - 1)^2 \geq 0$, the minimum value of $g(x)$ is $2$ (at $x^2 = 1$) and the maximum value is $\infty$. Thus, the range of $g(x)$ is $[2, \infty)$. Now, $f(x) = \log_{0.5}(g(x)) = \log_{1/2}(g(x)) = -\log_2(g(x))$. Since $g(x) \in [2, \infty)$, we have $\log_2(g(x)) \in [\log_2 2, \log_2 \infty) = [1, \infty)$. Multiplying by $-1$, we get $-\log_2(g(x)) \in (-\infty, -1]$. Therefore, the range of $f(x)$ is $(-\infty, -1]$.

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

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