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To find the inverse function of \( y = \sinh(x) = \frac{e^x - e^{-x}}{2} \), we need to solve for \( x \) in terms of \( y \).

1. Start with the original equation:
   \[
   y = \frac{e^x - e^{-x}}{2}
   \]

2. Multiply both sides by 2 to clear the fraction:
   \[
   2y = e^x - e^{-x}
   \]

3. Let \( u = e^x \), then \( e^{-x} = \frac{1}{u} \). The equation becomes:
   \[
   2y = u - \frac{1}{u}
   \]

4. Multiply through by \( u \) to get rid of the fraction:
   \[
   2yu = u^2 - 1
   \]

5. Rearrange the equation to form a quadratic equation in \( u \):
   \[
   u^2 - 2yu - 1 = 0
   \]

6. Solve this quadratic equation using the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -2y \), and \( c = -1 \):
   \[
   u = \frac{2y \pm \sqrt{(2y)^2 + 4}}{2}
   \]
   \[
   u = \frac{2y \pm \sqrt{4y^2 + 4}}{2}
   \]
   \[
   u = \frac{2y \pm 2\sqrt{y^2 + 1}}{2}
   \]
   \[
   u = y \pm \sqrt{y^2 + 1}
   \]

7. Since \( u = e^x \) must be positive, we choose the positive branch:
   \[
   e^x = y + \sqrt{y^2 + 1}
   \]

8. Take the natural logarithm of both sides to solve for \( x \):
   \[
   x = \ln\left(y + \sqrt{y^2 + 1}\right)
   \]

Thus, the inverse function is:
   \[
   x = \sinh^{-1}(y) = \ln\left(y + \sqrt{y^2 + 1}\right)
   \]

Would you like more details, or do you have any questions?

### Related Questions:
1. What are some properties of the hyperbolic sine function?
2. How do you find the inverse of other hyperbolic functions, such as \( \cosh(x) \) or \( \tanh(x) \)?
3. Can you explain why we only consider the positive solution in step 7?
4. What are some applications of inverse hyperbolic functions in calculus?
5. How can the Taylor series be used to approximate hyperbolic functions?

### Tip:
Inverse hyperbolic functions can often be used to solve integrals that involve hyperbolic functions, making them a useful tool in calculus.

求函数 y = shx = (e^x - e^-x)/2 的反函数

Learn how to find the inverse function of y = sinh(x) = (e^x - e^-x) / 2. This solution involves solving a quadratic equation and applying logarithmic properties to derive the final inverse. Ideal for advanced math students studying hyperbolic functions and their inverses.

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