Inverse hyperbolic functions

• Inverse hyperbolic functions come in several forms: sinh^(-1)x, cosh^(-1)x, and tanh^(-1)x. They are the inverses of the hyperbolic functions sinh x, cosh x, and tanh x respectively.

• These functions can be derived via point swapping from their equivalent hyperbolic function. In other words, if (a, b) is on the graph of the hyperbolic function, then (b, a) is on the graph of the corresponding inverse hyperbolic function.

• The function sinh is bijective, meaning it has a clear inverse that can be easily calculated as sinh^(-1)x = ln(x + sqrt{1 + x^2}).

• In contrast, cosh is only injective from [0, ∞) to [1, ∞), meaning cosh^(-1)x is defined only for x ≥ 1, and is calculated as cosh^(-1)x = ln(x + sqrt{x^2 - 1}).

• The third function, tanh, is also bijective and its inverse can be calculated as tanh^(-1)x = 0.5ln{(1 + x) / (1 - x)}.

• When working with inverse hyperbolic functions, the same trigonometric identities that work for normal hyperbolic functions apply.

• Derivatives of inverse hyperbolic functions are not as intuitive as their base functions, but can be derived using implicit differentiation. For instance, the derivative of sinh^(-1)x is 1/sqrt(1 + x^2), while the derivative of cosh^(-1)x is 1/sqrt(x^2 - 1) (for x > 1), and the derivative of tanh^(-1)x is 1/(1 - x^2).

• Just like the hyperbolic functions, inverse hyperbolic functions are also exponential in nature and hence their graphs grow extremely quickly.

• They play an integral role in solving some types of integrals and differential equations, especially when dealing with hyperbolic functions.

• As you revise, try to use the logarithmic definitions of these functions to solve problems, such as to find the derivative or to simplify complex equations. This can make it easier to spot relationships and patterns.

• It would be beneficial to practice solving problems that combine multiple types of functions (i.e., inverse hyperbolic, hyperbolic, trigonometric, exponential, logarithmic, etc.). This will help develop versatility in manipulating different function types.

• Developing a solid understanding and comfort with these functions will serve you well as you progress in your mathematical journey. Remember to consistently practice and review these concepts to keep them fresh.