Inverse Functions

1. Inverse Functions

2. Inverse Relations 22 4 22 4 The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the original is the range of the inverse; and vice versa) Ex: and are inverses because their input and output are switched. For instance:

3. Tables and Graphs of Inverses Switch x and y Orginal Inverse Switch x and y (0,25) (20,25) (16,18) (4,14) (18,16) (2,16) (0,10) (6,4) (16,2) (4,6) (14,4) (10,0) Although transformed, the graphs are identical y = x Line of Symmetry:

4. Inverse and Compositions In order for two functions to be inverses: AND

5. One-to-One Functions A function f(x) is one-to-one on a domain D if, for every value c, the equation f(x) = chas at most one solution for every x in D. Or, for every a and bin D: Theorems: • A function has an inverse function if and only if it is one-to-one. • If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function.

6. The Horizontal Line Test If a horizontal line intersects a curve more than once, it’s inverseis not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.

7. The Horizontal Line Test If a horizontal line intersects a curve more than once, it’s inverseis not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.

8. Example Without graphing, decide if the function below has an inverse function. If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function. See if the derivative is always one sign: Since the derivative is always negative, the inverse of f is a function.

9. Find the Inverse of a Function • Switch the x and y of the function whose inverse you desire. • Solve for y to get the Inverse function • Make sure that the domains and ranges of your inverse and original function match up.

10. Example Find the inverse of the following: Only Half Parabola Switch x and y Really y = Solve for y Restrict the Domain! Full Parabola (too much) x=3 Make sure to check with a table and graph on the calculator.

11. Logarithms v Exponentials

12. Definition of Logarithm The logarithm base a of b is the exponent you put on a to get b: i.e. Logs give you exponents! The logarithm to the base e, denoted lnx, is called the natural logarithm. a > 0 and b > 0

13. Logarithm and Exponential Forms Logarithm Form 5 = log2(32) Base Stays the Base Input Becomes Output Logs Give you Exponents 25 = 32 Exponential Form

14. Examples • Write each equation in exponential form • log125(25) = 2/3 • Log8(x) = 1/3 • Write each equation in logarithmic form • If 64 = 43 • If 1/27 = 3x 1252/3 = 25 81/3 = x log4(64) = 3 Log3(1/27) = x

15. Example Complete the table if a is a positive real number and: All Reals All Positive Reals All Positive Reals All Reals Yes Yes Yes Yes Always Up Always Down

16. The Change of Base Formula The following formula allows you to evaluate any valid logarithm statement: For a and b greater than 0 AND b≠1. Example: Evaluate

17. Solving Equations with theChange of Base Formula Solve: Isolate the base and power Change the exponential equation to an logarithm equation Use the Change of Base Formula

18. Properties of Logarithms For a>0, b>0, m>0, m≠1, and any real number n. Logarithm of 1: Logarithm of the base: Power Property: Product Property: Quotient Property:

19. Example 1 Condense the expression:

20. Example 2 Expand the expression:

21. Example 3 Solve the equation:

22. AP Reminders Do not forget the following relationships:

23. Inverse Trigonometry

24. Trigonometric Functions Each one of these trigonometric functions fail the horizontal line test, so they are not one-to-one. Therefore, there inverses are not functions. Cosecant Sine Secant Cosine Cotangent Tangent

25. Trigonometric Functions with Restricted Domains In order for their inverses to be functions, the domains of the trigonometric functions are restricted so that they become one-to-one. Cosecant Sine Secant Cosine Cotangent Tangent

26. Trigonometric Functions with Restricted Domains

27. Inverse Trigonometric Functions Csc-1 Sin-1 Sec-1 Cos-1 Cot-1 Tan-1

28. Inverse Trigonometric Functions

29. Alternate Names/Defintions for Inverse Trigonometric Functions Arccot is different because it is always positive but tan can be negative.

30. Example 1 Evaluate: This expression asks us to find the angle whose sine is ½. Remember the range of the inverse of sine is .

31. Example 2 Evaluate: This expression asks us to find the angle whose cosecant is -1 (or sine is -1). Remember the range of the inverse of cosecant is .

32. Example 3 Evaluate: The embedded expression asks us to find the angle whose sine is 1/3. Draw a picture (There are infinite varieties): It does not even matter what the angle is, we only need to find: Find the missing side length(s) Is the result positive or negative?

33. Example 4 Evaluate: The embedded expression asks us to find the angle whose cosine is -1/6. Draw a picture (There are infinite varieties): Ignore the negative for now. It does not even matter what the angle is, we only need to find: Find the missing side length(s) Is the result positive or negative?

34. Example 3 Evaluate: The embedded expression asks us to find the angle whose tangent is x. Draw a possible picture (There are infinite varieties): It does not even matter what the angle is, we only need to find: Is the result positive or negative? Find the missing side length(s)

35. White Board Challenge Evaluate without a calculator:

36. White Board Challenge Evaluate without a calculator:

37. White Board Challenge Evaluate without a calculator:

38. White Board Challenge Evaluate without a calculator:

39. White Board Challenge Evaluate without a calculator: