The inverse hyperbolic functions are the inverses of the hyperbolic sine ($\sinh x$), hyperbolic cosine ($\cosh x$), and hyperbolic tangent ($\tanh x$) functions. They are denoted as $\text{arsinh}(x)$, $\text{arcosh}(x)$, and $\text{artanh}(x)$, respectively. Here are their definitions and formulas:

1. Inverse Hyperbolic Sine ($\text{arsinh}(x)$)

The inverse hyperbolic sine function is defined as: $\text{arsinh}(x) = \ln\left(x + \sqrt{x^2 + 1}\right)$ where $\ln$ is the natural logarithm.

2. Inverse Hyperbolic Cosine ($\text{arcosh}(x)$)

The inverse hyperbolic cosine function is defined as: $\text{arcosh}(x) = \ln\left(x + \sqrt{x^2 - 1}\right)$ This function is only defined for $x \geq 1$.

3. Inverse Hyperbolic Tangent ($\text{artanh}(x)$)

The inverse hyperbolic tangent function is defined as: $\text{artanh}(x) = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$ This function is defined for $-1 < x < 1$.

These functions are useful in various areas of mathematics, including calculus, complex analysis, and solving certain types of differential equations.

Would you like further details on any of these functions, or do you have any specific questions?

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Here are five related questions to deepen your understanding:

1. How do the domains of the inverse hyperbolic functions differ from their corresponding hyperbolic functions?
2. Can you explain how the inverse hyperbolic functions relate to logarithms?
3. How are inverse hyperbolic functions used in solving differential equations?
4. What are the derivatives of the inverse hyperbolic sine, cosine, and tangent functions?
5. How do inverse hyperbolic functions compare to their trigonometric counterparts?

Tip: When solving problems involving inverse hyperbolic functions, it’s often helpful to convert the expression into its logarithmic form for simplification.