Inverse of Transcendental Functions 1- Inverse of Trigonometric Functions 2- Inverse of Exponential Functions 3- Inverse of Hyperbolic Functions

1- Inverse of Trigonometric Functions Since the trigonometric functions are not one-to-one, so they don’t have inverse functions. However, if we restrict their domains, then we may obtain one-to-one functions that have the same values as the trigonometric functions and that have inverse over these restricted domains. For example, the function is not one –to-one on its natural domain R. However, when the domain is restricted to the interval –π/2 to π/2, it becomes one-to-one.

Important Rules

Example Find the domain of Solution

Important Rules

Example Evaluate Solution

Notes

Important Rules

Proof

Example Evaluate the given inverse function Solution

2- Inverse Exponential Functions Every exponential function of the form is a one-to-one function. It therefore has an inverse function, which is called the logarithmic function with base a and is denoted by. Domain: Range:

The Natural Logarithmic Function The logarithm with base e is called the natural logarithm and has a special notation

Basic Properties of Natural Logarithmic Function

Example Solve the following equations for x Solution

Example Sketch the function Solution x y x y x=2 x y

3- Inverse Hyperbolic Functions The hyperbolic functions sinh x is one-to-one functions and so they have inverse functions denoted by

Proof (1)

Proof (3)

Important Rules