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

3. 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.

5. Important Rules

6. Example Find the domain of Solution

8. Important Rules

11. Example Evaluate Solution

12. Notes

13. Important Rules

14. Proof

15. Example Evaluate the given inverse function Solution

16. 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:

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

18. Basic Properties of Natural Logarithmic Function

19. Example Solve the following equations for x Solution

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

21. 3- Inverse Hyperbolic Functions The hyperbolic functions sinh x is one-to-one functions and so they have inverse functions denoted by

22. Proof (1)

23. Proof (3)

24. Important Rules