The given function is:

$f(x) = \frac{2x}{e^x - 1}$

To find the domain of the function $f(x)$, we need to identify all values of $x$ for which the function is defined.

Step-by-Step Solution:

1. Identify the Denominator:

   The function $f(x)$ has a denominator $e^x - 1$. The function will be undefined when the denominator is zero because division by zero is undefined.

2. Solve for $e^x - 1 = 0$:

   $e^x - 1 = 0$ $e^x = 1$

   The exponential equation $e^x = 1$ holds true when $x = 0$.

3. Determine the Domain:

   The function $f(x) = \frac{2x}{e^x - 1}$ is undefined when $x = 0$. For all other values of $x$, the function is defined because $e^x - 1 \neq 0$.

Conclusion:

The domain of the function $f(x) = \frac{2x}{e^x - 1}$ is all real numbers except $x = 0$. In interval notation, the domain is:

$(-\infty, 0) \cup (0, \infty)$

Would you like more details or have any questions?

Related Questions:

1. How can we find the range of $f(x) = \frac{2x}{e^x - 1}$?
2. What is the limit of $f(x) = \frac{2x}{e^x - 1}$ as $x$ approaches 0?
3. Can we simplify the expression $\frac{2x}{e^x - 1}$ further for specific values of $x$?
4. How does the function $f(x) = \frac{2x}{e^x - 1}$ behave as $x$ approaches infinity?
5. What are the critical points of $f(x) = \frac{2x}{e^x - 1}$, and how do they affect the function's graph?

Tip: To find the domain of a function, always check for values that could make the function undefined, such as division by zero or taking the square root of a negative number.