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Question #40337

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [-1, 1] \to \mathbb{R}$ be defined by $f(x) = \int_0^x \ln\left(t + \sqrt{t^2+1}\right) dt$ Which of the following represents the value of $f^{-1}\left( 1 - \sqrt{2} + \ln(1 + \sqrt{2}) \right)$ ?

Options

A.
1
B.
-1
C.
0
D.
\text{Does not exist}

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

Function Definition (The 'One-to-One' Constraint)|Bijectivity and Inverse|Even and Odd Function Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental condition for the existence of an inverse function: bijectivity.

The 'Lethal Collision' occurs because students often rush to algebraically evaluate the integral and solve $f(x) = y$ to find $f^{-1}(y)$ , completely ignoring the domain.

By defining the function as the integral of an odd function, $f(x)$ becomes an even function on a symmetric interval $[-1, 1]$ .

Even functions inherently violate the one-to-one (injectivity) constraint.

Thus, despite $x=1$ yielding the target value, the lack of global injectivity renders the inverse function non-existent.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Function Definition (The 'One-to-One' Constraint)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free