Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(2 f(x)+3 f\left(\frac{1}{x}\right)=x^{2}-1\), then \(f(x)\) is (A) a periodic function (B) an even function (C) an odd function

Step-by-Step Solution

Answer

f(x) is an odd function.

Step 1: Understanding the Function Behavior

We begin by understanding the symmetry or periodic nature of the function. We are given the equation \(2 f(x) + 3 f\left(\frac{1}{x}\right) = x^2 - 1\). We need to explore if \(f(x)\) exhibits symmetry such as even, odd, or periodic behavior.

Step 2: Test Even Function Conditions

A function \(f(x)\) is even if \(f(-x) = f(x)\) for all \(x\). Substitute \(-x\) in place of \(x\) in the given equation: \(2 f(-x) + 3 f\left(\frac{-1}{x}\right) = (-x)^2 - 1 = x^2 - 1\). This form will help check if the original equation transforms into itself, implying even symmetry.

Step 3: Test Odd Function Conditions

A function \(f(x)\) is odd if \(f(-x) = -f(x)\). Substitute \(-x\) in the original equation: \(2 f(-x) + 3 f\left(\frac{-1}{x}\right) = x^2 - 1\) will help us understand if \(f(x)\) can satisfy the conditions for an odd function.

Step 4: Analyze for Periodicity

To investigate periodicity, consider if there is a constant \(T\) such that \(f(x) = f(x + T)\) for all \(x\). The given equation does not have a straightforward repetitive pattern or terms that suggest a periodic behavior. Hence the periodicity is unlikely.

Step 5: Conclude Based on Tested Conditions

The condition for an odd function states that \(f(x)\) meets the requirement: if \( fig(-xig) = -fig(xig) \), which aligns the changes in symmetry through substitution. Therefore, given the transformations from substituting negative terms, \(f(x)\) possesses the characteristics of an odd function.

Key Concepts

Even Function

An even function is a type of function that exhibits symmetry about the y-axis. This means that the graph of an even function looks the same on the left of the y-axis as it does on the right. Mathematically, a function \( f(x) \) is considered even if it satisfies the condition:

• \( f(-x) = f(x) \) for every value of \( x \) in the domain of \( f \).

To determine if a function is even, you replace \( x \) with \( -x \) and simplify. If the resulting expression is identical to the original \( f(x) \), the function is even.

For example, if we have the function \( f(x) = x^2 + 2 \), substituting \( -x \) gives us \( (-x)^2 + 2 = x^2 + 2 \), which is exactly \( f(x) \). Hence, this function is even.
When dealing with exercises, testing evenness helps predict symmetry patterns in functions, especially in given equations.

Odd Function

An odd function has a different form of symmetry compared to even functions. It is symmetric with respect to the origin, meaning that if you rotate the graph 180 degrees around the origin, it remains unchanged. The defining characteristic of an odd function is:

• \( f(-x) = -f(x) \) for every value of \( x \) in the domain of \( f \).

To test if a function is odd, substitute \( x \) with \( -x \) and then take the negative of the entire function. If the result equals the original function, it is odd.

Consider the function \( f(x) = x^3 \). Substituting \( -x \) gives \( (-x)^3 = -x^3 \), which is \(-f(x) \). Therefore, \( f(x) = x^3 \) is an odd function.
Odd functions are particularly important when analyzing symmetrical properties and behaviors in various mathematical contexts.

Periodic Function

Periodic functions repeat their values in regular intervals or periods across the domain. This means that a periodic function satisfies the condition:

• \( f(x + T) = f(x) \) for all \( x \) and some positive constant \( T \).

The smallest positive interval \( T \) for which the function repeats itself is called the period. A classic example of a periodic function is the sine function, \( \sin(x) \), with a period of \( 2\pi \).

To determine if a function is periodic, look for a repeating pattern in its structure. Functions that have cosine, sine, or any such circular function elements are prime candidates for periodic behavior. For instance, \( \cos(x) \) repeats every \( 2\pi \), making it periodic with a period of \( 2\pi \).
Identifying periodicity is crucial for understanding how functions behave over different intervals and is a key feature in many areas of physics and engineering, like signal processing and waves.