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Question #40588

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8\}$ . A partition of $S$ into four disjoint subsets $A, B, C, D$ , each of size 2, is chosen. Let $s_A, s_B, s_C, s_D$ denote the sum of the elements in $A, B, C, D$ respectively. A function $f: [0, 4] \to \mathbb{R}$ is defined as: $f(x) = \begin{cases} 2x + s_A & 0 \le x < 1 \\ x^2 + s_B & 1 \le x < 2 \\ 3x + s_C & 2 \le x < 3 \\ 4x + s_D & 3 \le x \le 4 \end{cases}$ Find the number of ordered quadruplets of subsets $(A, B, C, D)$ such that there exists a continuous injective function $g: \mathbb{R} \to \mathbb{R}$ for which the composite function $h(x) = g(f(x))$ is continuous on $[0, 4]$ .

Options

A.
2
B.
3
C.
4
D.
6

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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

Division into Groups|Injective Mapping (One-One)|Composition Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of composition continuity, injectivity, and combinatorics (domain partitioning).

Students often incorrectly assume that a discontinuous inner function $f(x)$ can be 'fixed' by a continuous outer function $g(x)$ to produce a continuous composition $g(f(x))$ .

While true for some functions (like absolute value or periodic functions), the critical trap here is the injectivity of $g(x)$ .

An injective function strictly preserves distinct inputs, meaning $g(L) = g(R)$ implies $L = R$ .

Therefore, the inner function $f(x)$ must be globally continuous.

Recognizing this reduces a complex functional equation into a discrete system of equations, ultimately requiring a systematic combinatorial partition of the set.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Division into Groups
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free