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Question #40371

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{x^2 - 6x + 10}{x^2 - 6x + 13} + \sqrt{2 - |x-2| - |x-4|}$ . Find the range of $f(x)$ .

Options

A.
$\left[\frac{1}{4}, \frac{2}{5}\right]$
B.
$\left[\frac{1}{4}, 1\right)$
C.
$\left[\frac{2}{5}, 1\right)$
D.
$\left[\frac{1}{4}, \frac{1}{2}\right]$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Domain and Range Nuance|Completing the Square in Quadratics|Triangle Inequality Mind Games

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to synthesize domain constraints hidden within radical and modulus functions before performing range analysis.

The trap lies in the rational function.

By completing the square, the denominator becomes $(x-3)^2 + 4$ , which is strictly positive and appears to allow $x \in \mathbb{R}$ , leading to the incorrect range $[1/4, 1)$ .

However, the radical term invokes the Triangle Inequality ( $|a| + |b| \ge |a+b|$ ), forcing $|x-2| + |x-4| = 2$ .

This restricts the domain to the bounded interval $[2, 4]$ , fundamentally altering the range.

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