1. Warm up 1. Graph the following piecewise function:

3. Lesson 9-2 The Algebra of Functions Objective: To perform operations with functions. To evaluate composite functions

4. Operations with Functions Given f(x) and g(x):

5. Example Given f(x) = x – 4 and g(x) = x 2 – 4 Find a) (f + g)(x) b) (f – g)(x) c) (f g)(x) d)

6. Practice Given f(x) = 2x 2 and g(x) =x 2 -5x+6, find: (f+g)(x) (f-g)(x) (fg)(x)

7. Composite Functions Composite functions are functions that are formed from two functions f(x) and g(x) in which the output or result of one of the functions is used as the input to the other function. Notationally we express composite functions as In this case the result or output from g becomes the input to f.

8. Example 1 Given the composite function Replace g(x) with x+2 Replace the variable x in the f function with x+2 Expand

9. Problem 1 For the functions find

10. Evaluating Composite Functions Evaluating functions is done the same way. f(g(x)) Work with the inside function first. g(x) Evaluate that function for the given domain. Use that answer to evaluate the outside function. f(x)

11. Suppose that and find:

12. Warm up

13. Lesson 9-3 One to One and Inverse functions Objective: To determine if a function is one to one and be able to find the inverse of a function

14. One-to-One Function For y = f(x) to be a 1-1 function, each x corresponds to exactly one y, and each y corresponds to exactly one x. A 1-1 function f passes both the vertical and horizontal line tests.

15. S: Social Security function IS one-one Joe Samanth a Anna Ian Chelsea George 123456789 223456789 333456789 433456789 533456789 633456789 AmericansSSN

16. HORIZONTAL LINE TEST for a 1-1 Function The function y = f(x) is a one-to-one (1-1) function if no horizontal line intersects the graph at more than one point.

17. Do these graphs pass the horizontal line test?

19. Inverse Functions The inverse of a relation is the set of ordered pairs obtained by switching the coordinates of each ordered pair in the relation. The graph of the inverse function is the REFLECTION of the graph of the original. The inverse is denoted by

20. 123456789 223456789 333456789 433456789 533456789 633456789 SSN Joe Samantha Anna Ian Chelsea George Americans The inverse of the social security function

21. A function, f, has an inverse function, g, if and only if the function f is a one-to-one (1-1) function. Existence of an Inverse Function

22. A function, f, has an inverse function, g, if and only if f(g(x)) = x and g(f(x)) = x, for every x in domain of g and in the domain of f. Definition of an Inverse Function

23. Finding the inverse of a 1-1 function Step1: Write the equation in the form Step2: Interchange x and y. Step 3: Solve for y. Step 4: Write for y.

24. Find the inverse of Step1: Step2: Interchange x and y Step 3: Solve for y

25. = = = cont’d

26. Practice Find the inverse of: f(x)= 2x -3 f(x)= 3x-1

27. Verifying Inverse Functions To verify that 2 functions are inverse you must show that f(g(x)) = x and g(f(x)) = x

29. Practice Verify that the following functions are inverses of each other: and

30. Graphing the inverse Put the original graph in Y= [2 nd ] [DRAW][8] [VARS][►][1][1][ENTER]