Q41 / 160 · TS EAMCET 2023 (12 May Shift 1) · Mathematics

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

  • A
    $(-\infty, \infty)$
  • Correct answer
    B
    $(-\infty, -1]$
  • C
    $[-1, \infty]$
  • D
    $[-1, -1]$

Solution

$f(x)=\log _{0.5}\left(x^4-2 x^2+3\right)=-\log _2\left[\left(x^2-1\right)^2+2\right]$ $\begin{aligned} & 0 \leq\left(x^2-1\right)^2 < \infty \Rightarrow 2 \leq\left(x^2-1\right)^2+2 \leq \infty \\ & \log _2 2 \leq \log _2\left[\left(x^2-1\right)^2+2\right] < \log _2 \infty \\ & 1 \leq \log _2\left[\left(x^2-1\right)^2+2\right] < \infty \\ & -\infty < -\log _2\left(x^2-1\right)^2+2 \leq-1 \end{aligned}$ $\therefore \quad$ Range of $f(x)$ is $(-\infty,-1]$.

Hence, the correct answer is option: 'B'.

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