Problem 72 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f\) is a continuous function from \(R\) to \(R\) and \(f(f(a))=a\) for some \(a \in R\), then the equation \(f(x)=x\) has (A) no solution (B) exactly one solution (C) at most one solution (D) at least one solution

Step-by-Step Solution

Answer

The equation \(f(x) = x\) has at least one solution (Option D).

Step 1: Understanding the Problem

We are given a function \(f\) such that \(f\) is continuous and \(f(f(a)) = a\) for some \(a \in \mathbb{R}\). We need to determine if the equation \(f(x) = x\) has any solutions.

Step 2: Initial Observation

Since \(f(f(a)) = a\), we know that \(f\) has the property of being involutory at \(a\), meaning applying \(f\) twice brings us back to \(a\). This information is crucial to deduce the existence of fixed points.

Step 3: Fixed Point Property Analysis

Let's explore if \(f(x) = x\) has solutions. Given that \(f(f(a)) = a\), it's plausible to investigate if \(f(a) = a\). If \(f(a) = a\), then \(x = a\) is a solution to \(f(x) = x\).

Step 4: Application of Continuity

The function \(f\) is continuous. If \(f(f(a)) = a\), then by continuity, \(f(x) = x\) must have at least one solution due to the Intermediate Value Theorem, or by analyzing the conditions under which a change from \(f(f(a))=a\) to \(f(x) = x\) exists.

Step 5: Conclusion Based on Properties

Since \(f(f(a)) = a\), \(f\) has a fixed point property. Coupled with continuity, this ensures that there is at least one solution for \(f(x) = x\). Hence, option D is correct: "at least one solution."

Key Concepts

Involutory functions

An involutory function is a fascinating type of function where applying the function twice brings us back to where we started. To think of it simply, it is like pressing a switch twice to return to the original state of a device. When we say a function \( f \) is involutory, it means \( f(f(x)) = x \) for every \( x \) in its domain. In our exercise, the condition \( f(f(a)) = a \) tells us that \( f \) has involutory behavior at least at the point \( a \). This property can be quite useful in identifying other features of \( f \), like fixed points. Such functions quite naturally involve symmetries and are often found in mathematical puzzles and transformations. Additionally, involutions are not just about symmetry—they can reveal deeper insights about equations and their solutions.

Fixed points

A fixed point of a function \( f \) is a point \( x \) such that \( f(x) = x \). Think of it as a rest stop on a journey, where starting from that point and applying the function does not move you anywhere. In the context of our exercise, we are examining whether such a point exists for the function \( f \). Knowing that \( f(f(a)) = a \), it intuitively suggests that if \( f \) behaves nicely (e.g., is continuous), then it must possess at least one point where it remains unchanged; this is a fixed point. If any point \( a \) satisfies \( f(a) = a \), it confirms the existence of a fixed point. Fixed points are incredibly important as they represent equilibrium states in various systems, from mathematics to biology and even games in economics.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that guarantees the existence of solutions under certain conditions. It states that if a function \( f \) is continuous on a closed interval \([a, b]\) and if \( f(a) \) and \( f(b) \) have opposite signs, then there is at least one number \( c \) in the interval \([a, b]\) such that \( f(c) = 0 \). This idea can be extended to finding fixed points. In our exercise, the continuity of \( f \) allows us to leverage the IVT. If there exists a sign change in \( f(x) - x \) over an interval, by the IVT, there must be a point where \( f(x) - x = 0 \), which is the same as finding \( f(x) = x \). Thus, the IVT solidly supports the conclusion that there must be at least one solution to the equation, confirming at least one fixed point for the continuous involutory function \( f \). It’s a powerful tool that turns the abstract concept of continuity into practical assurance of solutions.