Q30 / 75 · JEE Main 2025 (28 Jan Shift 2) · Mathematics

Let $[x]$ denote the greatest integer less than or equal to $x$. Then the domain of $f(x)=\sec ^{-1}(2[x]+1)$ is :

  • A
    $(-\infty,-1] \cup[0, \infty)$
  • B
    $(-\infty,-1] \cup[1, \infty)$
  • Correct answer
    C
    $(-\infty, \infty)$
  • D
    $(-\infty, \infty)-\{0\}$

Solution

$\begin{array}{ll} f(x)=\sec ^{-1}(2[x]+1) \\ \Rightarrow 2[x]+1 \geq 1 & \text { or } 2[x]+1 \leq-1 \\ \Rightarrow 2[x] \geq 0 & \text { or } 2[x] \leq-2 \\ \Rightarrow[x] \geq 0 & \text { or }[x] \leq-1 \\ \Rightarrow x \geq 0 & \text { or } x \leq 0 \end{array}$ Domain of $f(x)$ is $(-\infty, \infty)$

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

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