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Question #40331

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{1}{\sqrt{[x]^2 - 3[x] + 2}}$ and $g(x) = \frac{12}{x^2 + 2x + 5}$ , where $[ \cdot ]$ denotes the greatest integer function. Let $D$ be the domain of the composite function $h(x) = f(g(x))$ . Find the sum of all integers that do NOT belong to $D$ .

Options

A.
-5
B.
-4
C.
-2
D.
0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Domain and Range Nuance|Composition Continuity|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

The question tests the nuanced evaluation of step function inequalities composed with bounded functions.

The primary trap lies in solving $[g(x)]^2 - 3[g(x)] + 2 > 0$ .

Students must deduce $[g(x)] \le 0$ or $[g(x)] \ge 3$ .

A common mistake is to drop the step function and solve $g(x) > 2$ , leading to an incorrect domain and a sum of $-2$ .

Another lethal trap occurs when students correctly find $[g(x)] \ge 3$ but erroneously conclude it has no solution because the maximum value of $g(x)$ is exactly $3$ , forgetting that $[3] = 3 \ge 3$ .

This causes them to miss the isolated point $x = -1$ in the domain, leading them to incorrectly include $-1$ in the excluded integers and arrive at the trap answer of $-5$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Domain and Range Nuance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free