1. Composite and Inverse Functions Review and additional information on sections 1.8 and 1.9

2. Example of Composition of Functions  Let f(x) = x 2 + 2x and g(x) = 3x - 1  evaluate (f  g)(2)  (f  g)(x) = f(g(x)) = f(3x – 1) = (3x – 1) 2  (f  g)(x) = f(g(x)) = f(3x – 1) = (3x – 1) 2 + 2(3x – 1)  (f  g)(x) = (3x – 1) ( 3x – 1) + 6x – 2  (f  g)(x) = 9x 2 – 3x – 3x + 1 + 6x – 2  (f  g)(x) = 9x 2 – 1  So (f  g)(2) = 9(2) 2 – 1 = 35

3. How to Evaluate Combining of Functions Numerically?  Given numerical representations for f and g in the table  Find f(g(5)), f(g(6)), f(g(7)), f(g(8)) f(g(5))=7f(g(6))=8f(g(7))=5f(g(8))=6

4. How to Evaluate Combining of Functions Graphically? Use graph of f and g below to evaluate (f  g) (1) y = g(x) y = f(x) Can you identify the two functions? Try to evaluate them now. Hint: Look at the y-value when x = 1.

5. How to Evaluate Combining of Functions Graphically? y = g(x) y = f(x)  (f  g) (1) = f(g(1)) = f(0) = 2 Check your answer now.

6. Inverse functions - one function undoes the other. x f(x) 0 -3 1 -1 2 1 3 x g(x) -3 0 -1 1 1 2 3 3 Definition of Inverse Functions - If functions f and g are such that for all x in the domains of f and g, then the f and g functions are said to be inverses of each other.

7. Finding an inverse function: 1. Rewrite the function in y= form 2. Switch x and y 3. Solve for y 4. Adjust the domain of the original function if necessary 5. The new function is

8. Find the inverse of each function.

9. A One to One function A function is one to one if exactly one element in the domain is paired with exactly one element in the range.

10. Examples of one to one functions 1.A person and his or her id number 2.A person and his or her passport number 3.Any function whose graph passes the horizontal test

11. Examples of one to one functions Note every Uppercase number has exactly one lower case partner ABCABC ABCABC abcabc

12. Examples of one to one functions 1. 2. f (x) = 3x +5 3.

13. Examples of functions that are not one to one 1.f (x) = |x| 2. 3.

14. Examples  Domain x Range 3x -2-6 -3 00 13 26 39 Domain x -6-2 -3 00 31 62 93

15. One-to-one functions- all x values have unique y answers and If a function is one-to-one its inverse will also be a function without limiting the domain. A function is one-to-one if it passes the horizontal line test.

16. The inverse of a function is written

17. Inverse Functions  Are and  2( x - 3) +6 x – 6 + 6 = x  (2x + 6) – 3  x + 3 – 3= x

18. Graphs of Inverse Functions Note inverse functions are reflected about the line y = x