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Question #41128

Difficulty: 8/10 | 🟢 Free

The Question

Consider the function $f: [-1, \infty) \to \left[-\frac{1}{3}, \infty\right)$ defined by $f(x) = \frac{x^3}{3} + 2x^2 + 3x + \sum_{n=1}^{\infty} \frac{x^2}{(1+x^2)^n}$ Which of the following statements is true?

Options

A.
\text{Bijective}
B.
\text{Injective but not surjective}
C.
\text{Surjective but not injective}
D.
\text{Neither injective nor surjective}

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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

Bijectivity and Inverse|Sum to n terms vs. General Term|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of infinite series convergence, the Method of Intervals (Wavy Curve), and global bijectivity.

The primary trap lies in the piecewise nature of the infinite geometric series: students often evaluate the sum to 1 and assume f(x) is a continuous polynomial.

By applying the Wavy Curve method to f'(x) = (x+1)(x+3), they correctly identify that the derivative is positive on [-1, infinity) and incorrectly conclude the function is strictly increasing and therefore bijective.

However, the sum evaluates to 0 at x=0, creating a point discontinuity.

This discontinuity causes the function to map a negative value to 0 while f(0)=0 (destroying injectivity) and skips the value 1 entirely (destroying surjectivity).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Bijectivity and Inverse
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free