Problem 22 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f(x)\) is an odd periodic function with period 2 , then \(f(4)\) equals (A) \(-4\) (B) 4 (C) 2 (D) 0

Step-by-Step Solution

Answer

The answer is (D) 0.

Step 1: Understand Odd Functions

An odd function has the property that for any value of \(x\), \(f(-x) = -f(x)\). This means if you take the function's value at any point and flip the sign and input, you should get the negative of the original function's value.

Step 2: Understand Periodic Functions

A periodic function with period 2 means that \(f(x) = f(x+2)\) for every \(x\). This implies every 2 units along the x-axis, the function repeats itself.

Step 3: Determine the Function Value

Since \(f(x)\) is a periodic function with period 2, \(f(4) = f(4-2) = f(2)\). Using the property of odd functions, \(f(2) = -f(-2)\). But, due to periodicity, \(f(-2) = f(0)\), so \(f(2) = -f(0)\).

Step 4: Use Odd Function Property

Since \(f(0)\) for an odd function must be 0 (because \(f(-0) = -f(0)\) giving \(f(0) = -f(0)\), hence \(f(0) = 0\)), \(f(2) = -0 = 0\). Thus, \(f(4) = f(2) = 0\).

Key Concepts

Properties of Odd Functions

Odd functions have a distinct symmetry about the origin in the coordinate plane. This means if you were to fold the graph along the line \(y=x\), the left side would be a mirror image of the right side, but flipped upside down. You can express this mathematical property with the equation \(f(-x) = -f(x)\).

• This takes any point \((x, f(x))\) and ensures the point \((-x, -f(x))\) also exists.
• It means the function is symmetrical with respect to the origin, often referred to as "origin symmetry."

At the heart of this symmetry is the idea that if the input \(x\) changes its sign, the output is the negative of the original. Therefore, verifying if a function is odd requires checking this relationship for various values.

Properties of Periodic Functions

Periodic functions are all about repetition at regular intervals along their domain. This characteristic is particularly useful in the contexts where repeating patterns are important, like in signal processing, sound waves, and seasonal trends. A periodic function can be identified by the equation \(f(x) = f(x+T)\), where \(T\) is the period.

• This means after every \(T\) distance along the x-axis, the function values repeat.
• The smallest possible positive \(T\) that satisfies this is called the fundamental period.

Take a function with period 2, for example. It indicates that the entire graph repeats itself over every interval of length 2. Thus, evaluating these functions just takes a matter of finding where you are in the current cycle.

Function Evaluation

Evaluating functions, especially ones with both symmetry and periodicity, can be simplified through their unique properties. With odd periodic functions, these evaluations often become straightforward using their characteristic equations. When solving for a value like \(f(4)\), leverage both properties:

• Use periodicity to revert back to the starting section by reducing using its period: since 4 is two periods into the repetition cycle, you calculate \(f(4)\) as \(f(4-2)=f(2)\).
• Apply the odd function property: if \(f(2) = -f(-2)\) and through periodicity \(f(-2) = f(0)\), then \(f(2)\) must be \(-f(0)\).

Knowing \(f(0)\) is zero for an odd function helps quickly evaluate \(f(2)\) to zero, and consequently \(f(4)\) is also zero. Therefore, understanding these properties not only simplifies calculations but also eliminates guesswork in finding exact values.