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The problem asks to find the inverse of the function:

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

This is a common hyperbolic function known as the hyperbolic sine function, written as:

\[
y = \sinh(x)
\]

The inverse of the hyperbolic sine function is called the **inverse hyperbolic sine** or **arsinh** (or \(\text{arsinh}(y)\)):

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

Thus, the inverse of the given function is:

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

Let me know if you would like further details on this! Here are some related questions to explore:

1. How do you prove that the inverse of \(\sinh(x)\) is \(\ln(y + \sqrt{y^2 + 1})\)?
2. What are the properties of hyperbolic functions?
3. How is the inverse of the hyperbolic cosine function (\(\cosh^{-1}\)) derived?
4. How do you differentiate and integrate hyperbolic functions?
5. Can you provide more examples of inverse hyperbolic functions?

**Tip:** Hyperbolic functions like \(\sinh(x)\) and \(\cosh(x)\) are analogues of trigonometric functions but for a hyperbola instead of a circle.

求函数 y = (e^x - e^(-x)) / 2 的反函数。

Find the inverse of the hyperbolic sine function y = (e^x - e^(-x)) / 2. Learn about hyperbolic functions, their inverses, and how to derive the formula for arsinh(y). Suitable for advanced high school students in Grades 11-12.

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