https://r2.erweima.ai/mathbot/image/1726362393983osx17cq2.jpg

The equation provided in the image is:

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

This is the definition of the hyperbolic sine function, also written as:

\[
y = \sinh(x)
\]

The problem asks to find the inverse function of \( y = \sinh(x) \). The inverse of the hyperbolic sine function is known as the **inverse hyperbolic sine**, or **arsinh** (also denoted \( \text{arsinh}(x) \) or \( \sinh^{-1}(x) \)).

The inverse of \( y = \sinh(x) \) is:

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

Thus, the inverse function is:

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

Let me know if you would like further explanation or have any questions!

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Here are 5 related questions:
1. What is the inverse function of \( \cosh(x) \) and how is it derived?
2. How does the hyperbolic sine function relate to the exponential function?
3. What are the properties of the inverse hyperbolic sine function?
4. How do we differentiate and integrate hyperbolic sine and its inverse?
5. Can you explain the geometric interpretation of hyperbolic functions?

**Tip:** The hyperbolic functions \(\sinh(x)\), \(\cosh(x)\), and others behave similarly to trigonometric functions but are defined using exponentials. Understanding their inverses can help in solving complex integrals and differential equations.

求函数 y = (e^x - e^(-x)) / 2 的反函数。

Discover how to find the inverse function of y = (e^x - e^(-x)) / 2, also known as the hyperbolic sine function. This detailed explanation includes the formula for the inverse hyperbolic sine and provides an understanding of the concept of inverse functions in hyperbolic functions. Suitable for university-level students.

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