Proving the Odd Function Property of a Given Exponential Function

When dealing with properties of functions in calculus and mathematical analysis, one fundamental property is whether a function is odd or even. An odd function satisfies the condition f(-x) -f(x). This article will delve into the proof of this property for the given function, f(x) -x - frac{e^{-x} - 1}{e^{-x} 1}.

Understanding Odd Functions

First, let's revisit the concept of an odd function. A function f(x) is considered odd if for every point x in its domain, the following holds true:

f(-x) -f(x)

Step 1: Calculate f(-x)

Let's start by calculating f(-x) for the given function:

f(x) -x - frac{e^{-x} - 1}{e^{-x} 1}

First, substitute -x into the function:

f(-x) -(-x) - frac{e^{-(-x)} - 1}{e^{-(-x)} 1}

Since e^{-x} frac{1}{e^x}, we have:

f(-x) x - frac{e^x - 1}{e^x 1}

Step 2: Simplify f(-x)

Now, we need to simplify the expression:

x - frac{e^x - 1}{e^x 1}

To simplify, let's combine the terms into a single fraction:

x - frac{e^x - 1}{e^x 1} frac{x(e^x 1) - (e^x - 1)}{e^x 1}

Distribute x in the numerator:

frac{x(e^x 1) - (e^x - 1)}{e^x 1} frac{xe^x x - e^x 1}{e^x 1}

Combine like terms in the numerator:

frac{xe^x x - e^x 1}{e^x 1} frac{(xe^x - e^x) (x 1)}{e^x 1}

Factor out e^x in the first term of the numerator:

frac{e^x(x - 1) (x 1)}{e^x 1}

This form is still complex, so let's break it down further:

frac{e^x(x - 1) (x 1)}{e^x 1}

Since (e^x 1) e^x 1, we can rewrite:

frac{e^x(x - 1) (x 1)}{e^x 1} -frac{e^x - 1}{e^x 1} x

Therefore:

f(-x) -left(x - frac{e^x - 1}{e^x 1}right)

This can be simplified as:

f(-x) -f(x)

Conclusion

The given function f(x) -x - frac{e^{-x} - 1}{e^{-x} 1} is indeed an odd function, as it satisfies the condition f(-x) -f(x).

Additional Properties and Applications

Understanding whether a function is odd or even is crucial in various mathematical applications, including Fourier series, symmetry analysis, and solving differential equations. Odd functions have symmetry about the origin, and even functions have symmetry about the y-axis. This property plays a significant role in simplifying complex expressions and solving integrals.

Further Exploration

For further exploration, one can investigate the derivatives and integrals of such functions, as these operations preserve the odd/even nature of the function. Additionally, studying the behavior of these functions under transformations (such as scaling, shifting, and combining) provides deeper insights into the nature of mathematical functions in a broader context.