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JEE Main 202528 JanuaryEvening Shift
Exam
JEE Main
Year
2025
Date
28 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Functions

JEE Main 2025 · 28 January · Evening Shift · Mathematics · Functions

Let [x] denote the greatest integer less than or equal to x . Then...

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

Options

  1. A(−∞,−1]∪[0,∞)(-\infty,-1] \cup[0, \infty)
  2. B(−∞,−1]∪[1,∞)(-\infty,-1] \cup[1, \infty)
  3. C(−∞,∞)(-\infty, \infty)
  4. D(−∞,∞)−{0}(-\infty, \infty)-\{0\}

Correct answer

C. (−∞,∞)(-\infty, \infty)

Step-by-step solution

f(x)=sec⁡−1(2[x]+1)⇒2[x]+1≥1 or 2[x]+1≤−1⇒2[x]≥0 or 2[x]≤−2⇒[x]≥0 or [x]≤−1⇒x≥0 or x≤0\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)f(x) is (−∞,∞)(-\infty, \infty)
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