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Question #41154

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = x[x]$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Let $S$ be the range of $f$ . Which of the following statements is TRUE?

Options

A.
$f$ is injective on $\mathbb{R} \setminus [0, 1)$ and $S = [0, \infty)$
B.
$f$ is not injective, and $S = [0, \infty) \setminus \{n^2+n \mid n \in \mathbb{N}\}$
C.
$f$ is bijective on $[1, \infty)$ with inverse $f^{-1}(x) = \frac{x}{[\sqrt{x}]}$
D.
$f$ is constant on $[0, 1)$ , and $S = [0, \infty) \setminus \{n^2 \mid n \in \mathbb{N}\}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Greatest Integer Function Traps|Injective Mapping (One-One)|Bijectivity and Inverse

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the global injectivity and range of a function involving the Greatest Integer Function.

The primary trap is assuming that because the algebraic structure $x[x]$ is strictly increasing on positive intervals, it maps continuously to $[0, \infty)$ .

Students often fail to check negative intervals which map to positive values, violating injectivity ( $f(n)=f(-n)$ ), and they overlook the constant step-function behavior on $[0, 1)$ .

Furthermore, when finding the range, students incorrectly assume it is a continuous set, missing the discrete 'holes' at $n^2+n$ caused by the open boundaries of the step-wise domain intervals.

Option C is a strong distractor because the formula for the inverse is actually correct on the restricted range, but the function is not surjective on $[1, \infty)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Greatest Integer Function Traps
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free