Problem 22 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)\) is an odd periodic function with period 2 , then \(f(4)\) equals (A) \(-4\) \(\begin{array}{lll}\text { (B) } 4 & \text { (C) } 2 & \text { (D) } 0\end{array}\)

Step-by-Step Solution

Answer

The answer is 0; thus, (D) is correct.

Step 1: Understanding an Odd Function

An odd function satisfies the property \(f(-x) = -f(x)\). This means that the function is symmetric with respect to the origin.

Step 2: Understanding a Periodic Function

A periodic function with period 2 means that \(f(x) = f(x + 2)\) for any \(x\). The function repeats its values every interval of 2.

Step 3: Expressing f(4) Using Periodicity

Since \(f(x) = f(x + 2)\), we apply the periodicity to find \(f(4)\). Starting from \(f(0)\), we have \(f(2) = f(0)\) and thus \(f(4) = f(0)\) again, repeating the values every 2 units.

Step 4: Analyzing Odd Property at Integer k

To find the value at integers, especially \(f(0)\), consider the oddness of a function. Given that \(f(x) = -f(-x)\), if \(f(x)\) is an odd function, then \(f(0) = -f(0)\). This implies that \(f(0)=0\), as the only number equal to its own negative is zero.

Step 5: Conclusion

Since \(f(4) = f(0)\) due to periodicity and \(f(0) = 0\) because it is odd, we conclude that \(f(4) = 0\). Therefore, the correct answer is option (D) 0.

Key Concepts

Odd Function Properties

An odd function is an important concept in mathematics, characterized by specific symmetry. To understand odd functions, it is crucial to know that they follow a general property defined mathematically as \( f(-x) = -f(x) \). This property means that the graph of an odd function has origin symmetry. This is a special kind of symmetry where the graph looks the same when you rotate it 180 degrees around the origin.
For example, the function \( f(x) = x^3 \) is odd because when you replace \( x \) with \( -x \), it becomes \( (-x)^3 = -(x^3) \). This exactly follows the rule for odd functions, creating a mirror image in all four quadrants when reflected through the origin.
Some key aspects of odd functions include:

• The function always passes through the origin, meaning \( f(0) = 0 \), as it's the only way for the equation \( f(-0) = -f(0) \) to hold true.
• Odd functions can never have a vertical or horizontal reflection symmetry, because their symmetry is rotational.

Recognizing these properties helps in identifying odd functions and applying their unique characteristics in problem solving.

Periodic Function Properties

Periodic functions are functions that repeat their values at regular intervals. The most distinctive characteristic of such functions is defined by the concept of the period. The period of a function is the smallest positive number \( p \) such that \( f(x) = f(x + p) \) for all values of \( x \).
A familiar example of a periodic function is the sine function, \( f(x) = \sin x \), which repeats every \( 2\pi \).
For any periodic function with a given period, certain properties hold:

• The function's entire graph over one period can be replicated infinitely along the x-axis.
• All periods beyond the first do not change the essence of the function; they simply continue the pattern.
• In the context of this exercise, a period of 2 implies that the function's values repeat every 2 units along the x-axis, which provides predictability and a pattern in its behavior.

Understanding these properties helps in solving problems where the goal is to find function values at specific points.

Mathematical Symmetry

Symmetry in mathematics refers to the balanced proportions and harmony within the graph of a function. Symmetrical properties help in predicting the behavior and properties of functions without always having to compute specific values.
There are different types of symmetry a function can exhibit:

• **Origin Symmetry**: As seen in odd functions, origin symmetry means the function is symmetric with respect to the point \((0, 0)\). This type of symmetry results in a graph that looks the same when rotated 180 degrees about the origin.
• **Axis Symmetry**: Even functions exhibit symmetry along the y-axis, meaning the graph is a mirror image on either side of the axis.

In this exercise, the concept of symmetry aids in understanding how the function behaves as both an odd and periodic function, making it a powerful tool in solving equations efficiently.
Symmetry simplifies problems and reduces the complexity involved in determining function values, as it allows for logical inferences based on repetitive or reflected patterns.