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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.

Discover how to find the domain of the function f(x) = 2x / (e^x - 1) step-by-step. Learn about exponential functions, division by zero, and function definitions. Suitable for high school students and anyone interested in calculus and functions.

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