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The given equation is:

\[
y = \frac{e^x - e^{-x}}{2}
\]

This function is known as the **hyperbolic sine function**, commonly written as:

\[
y = \sinh(x)
\]

We are tasked with finding the inverse of this function, which is called the **inverse hyperbolic sine**, denoted as \( \text{arsinh}(y) \) or \( \sinh^{-1}(y) \).

The inverse of the hyperbolic sine function is given by the formula:

\[
x = \sinh^{-1}(y) = \ln(y + \sqrt{y^2 + 1})
\]

Thus, the inverse function is:

\[
x = \ln(y + \sqrt{y^2 + 1})
\]

Would you like to see a step-by-step explanation of how this inverse function is derived?

Here are 5 related questions you might be interested in:
1. How do you derive the formula for the inverse hyperbolic sine function?
2. What are the properties of hyperbolic sine functions?
3. Can the inverse function be applied to solve real-world problems?
4. How do hyperbolic functions relate to trigonometric functions?
5. How do you differentiate and integrate hyperbolic functions?

**Tip**: The hyperbolic sine and cosine functions, while similar to their trigonometric counterparts, are used in different contexts such as modeling hyperbolas and in calculus of complex functions.

求函数 y = (e^x - e^(-x)) / 2 的反函数。

Learn how to find the inverse of the hyperbolic sine function, y = (e^x - e^(-x)) / 2. This problem explores key mathematical concepts, including hyperbolic functions and inverse functions. Suitable for advanced high school students in Grades 11-12.

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