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Question #40593

Difficulty: 8/10 | 🟢 Free

The Question

Consider the set of functions $S = \{f_1, f_2, f_3, f_4, f_5\}$ from $\mathbb{R}$ to $\mathbb{R}$ defined as: $f_1(x) = -e^x, \quad f_2(x) = x^3 + x, \quad f_3(x) = x^4 + x^2$ $f_4(x) = -x^5 - x, \quad f_5(x) = \ln(e^x + 1)$ A non-empty subset of $S$ is chosen and its elements are composed in decreasing order of their indices to form a new function $H(x)$ . (For example, if $\{f_1, f_3, f_4\}$ is chosen, $H(x) = f_4(f_3(f_1(x)))$ ). Find the number of such subsets for which $H(x)$ is strictly increasing on $\mathbb{R}$ .

Options

A.
11
B.
13
C.
15
D.
16

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

Total Selection (at least one)|Composite Injectivity/Surjectivity|Compound Function Boundaries|Increasing/Decreasing Strictness

Why This Question Is Tricky

Common Mistakes

This question tests the 'Compound Function Boundary' trap, where students incorrectly assume that a composition involving a non-monotonic function (like $f_3(x) = x^4+x^2$ ) will always result in a non-monotonic function.

The trap forces students to realize that if the inner function's range is restricted to a domain where the outer function is strictly monotonic (e.g., $f_1(x)$ restricts the range to $\mathbb{R}^-$ , where $f_3$ is strictly decreasing), the overall composition becomes strictly monotonic.

Furthermore, it tests the strict application of the chain rule for monotonicity ( $f'(g(x)) \cdot g'(x) > 0$ ) across multiple compositions, requiring a systematic state-machine approach to track both monotonicity and range signs.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Total Selection (at least one)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free