f\left(x\right)\ =tanh^{-1}\left(tan\left(x\right)\right)

The function you've provided is:

\[
f(x) = \tanh^{-1}\left(\tan(x)\right)
\]

### Understanding the Function

1. **Inverse Hyperbolic Tangent**: The inverse hyperbolic tangent function, \(\tanh^{-1}(y)\), is defined as:
   \[
   \tanh^{-1}(y) = \frac{1}{2} \ln\left(\frac{1 + y}{1 - y}\right)
   \]
   provided that \( |y| < 1 \).

2. **Substitute the Expression**: In your case, \( y = \tan(x) \), so:
   \[
   f(x) = \frac{1}{2} \ln\left(\frac{1 + \tan(x)}{1 - \tan(x)}\right)
   \]

### Domain Considerations

- For \(\tanh^{-1}(y)\) to be defined, \( |y| < 1 \). Therefore, \( |\tan(x)| < 1 \).

- The tangent function, \(\tan(x)\), has asymptotes where it becomes undefined, specifically at:
  \[
  x = \frac{\pi}{2} + n\pi \quad \text{for integers } n.
  \]

  Between these asymptotes, \(\tan(x)\) ranges from \(-\infty\) to \(+\infty\). Thus, the condition \( |\tan(x)| < 1 \) is satisfied when \( x \) is within the intervals:
  \[
  \left( -\frac{\pi}{4} + n\pi, \frac{\pi}{4} + n\pi \right)
  \]
  for integers \( n \).

### Simplified Function

Thus, the function can be expressed as:
\[
f(x) = \frac{1}{2} \ln\left(\frac{1 + \tan(x)}{1 - \tan(x)}\right)
\]
within the domain where \( |\tan(x)| < 1 \).

Would you like more details or have any questions about this function?

Here are 5 related questions:
1. What is the derivative of \( f(x) = \tanh^{-1}\left(\tan(x)\right) \) with respect to \( x \)?
2. What is the range of the function \( f(x) \)?
3. How does the function \( f(x) \) behave near the boundaries of its domain?
4. Can \( f(x) \) be expressed using trigonometric identities for easier analysis?
5. What are the applications of the function \( f(x) = \tanh^{-1}\left(\tan(x)\right) \) in physics or engineering?

**Tip**: Always check the domain of the inner function when dealing with composite functions, especially when involving inverse functions, to avoid undefined or complex values.

Explore the function f(x) = tanh^{-1}(tan(x)) with a comprehensive explanation. Learn about inverse hyperbolic tangent, domain considerations, and simplified forms of the function. Suitable for advanced high school students and beyond.

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