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Identifying the Odd One Out- Unveiling the Odd Function Among the Options

Which of the following is an odd function? This question often arises in mathematics, particularly in the study of functions and their properties. Understanding odd functions is crucial for various applications in physics, engineering, and other scientific disciplines. In this article, we will explore the concept of odd functions, their characteristics, and how to identify them among a set of given functions.

Odd functions are a special type of function that exhibit a unique property: they are symmetric about the origin. In other words, if a function f(x) is odd, then f(-x) = -f(x) for all x in the function’s domain. This property makes odd functions distinct from even functions, which are symmetric about the y-axis.

To determine whether a given function is odd, we can apply the following steps:

1. Substitute -x for x in the function.
2. Simplify the resulting expression.
3. Compare the simplified expression with the original function.

If the simplified expression is equal to the negative of the original function, then the function is odd. Let’s examine some examples to illustrate this process.

Example 1: f(x) = x^3

Substituting -x for x, we get f(-x) = (-x)^3 = -x^3.

Since f(-x) = -f(x), we can conclude that f(x) = x^3 is an odd function.

Example 2: f(x) = x^2

Substituting -x for x, we get f(-x) = (-x)^2 = x^2.

Since f(-x) = f(x), we can conclude that f(x) = x^2 is an even function, not an odd function.

Example 3: f(x) = x^3 + 3x

Substituting -x for x, we get f(-x) = (-x)^3 + 3(-x) = -x^3 – 3x.

Since f(-x) = -f(x), we can conclude that f(x) = x^3 + 3x is an odd function.

In summary, to identify an odd function, we can follow the steps outlined above. Recognizing odd functions is essential for understanding their properties and applications in various fields. By analyzing the behavior of odd functions, we can gain insights into the underlying mathematical principles and their practical implications.

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