1. SWBAT… define a function, learn function notation, and a evaluate function Agenda 1. Cell phone problem (10 min) 2. Lots of notes on functions with many practice problems (25 min) 3. OYO Problems (10 min) Warm-Up: 1. Set up your notes – Topic is “Functions” HW#5: Functions – Page 1 Wed, 10/31 Happy Halloween!

2. Objectives Today: 1. To define a function 2. To learn function notation 3. To evaluate functions Tomorrow: 1. To learn function mapping 2. To conduct the vertical line test 3. To find the domain and range of a function 4. To write a function as an ordered pair

3. Ms. Sophia Papaefthimiou Infinity HS

4. What is a function? A function is like a machine: it has an input and an output.

5. Function Notation The most common name is "f", but you can have other names like "g" What goes into the function (input) is put inside parentheses after the name of the function Example: f(x) (pronounced “f of x”) The function is called "f“ "x" goes in (input) Question: A function is called “g” and “a” is the input. Write the function.

6. Output Value Range Dependent Variable Input Value Domain Independent Variable Name of the function Function Notation: The Symbolic Form

7. Function Notation Function notation replaces the ___ in an equation with ___ Example: Given y = 3x + 2, write the equation in function notation f(x) = 3x + 2 Question: Write y = x 2 in function notation. f(x) = x 2 y f(x)

8. Function Notation You used to say “y = 2x + 3; find the value of y when x = -1” y = 2x + 3 y = 2(-1) + 3 y = -2 + 3 y = 1 Now you say “f(x) = 2x + 3; find f(-1)” 1 f(-1) = 2x + 3

9. Ex: If f(x) = 2x + 3 find: 1. f(4x) 2. f(1) + f(9) 3. f(t – 5)

10. OYO (On Your Own) Problems Directions: If f(x) = 2x – 4 and g(x) = x² – 4x, find each value: 1. f(-3) 2. f(3x) (Hint: Replace the x with 3x) 3. g(t) (Hint: Use the g function, replace x with t) 4. f(q + 1) (Hint: f(q+1) = 2(q+1) – 4 5. f(2) + g(-2) 6. f(g(-2)) (Hint: Start from the inside out. Find g(-2) first) Then find f(12)

11. Revisit our objectives Today: 1. To define a function 2. To learn function notation 3. To evaluate functions

12. Agenda 1. Warm-Up (10 min) 2. Notes on functions (30 min) Warm-Up: If f(x) = -3x + 5 and g(x) = x² – 6, find: 1.) f(-8) 2.) g(-4) 3.) f(2) – g(1) 4.) f(g(-4)) (Hint: Start from the inside out. Find g(-4) first) Then find f(10)) HW#3: Evaluating functions (page 1) SWBAT… define and evaluate functions Fri, 11/2

14. A function P is defined as follows: For x > 0, P(x) = x 5 + x 4 – 36x – 36 For x < 0, P(x) = -x 5 + x 4 + 36x – 36 What is the value of P(-1)? A. -70 B. -36 C. 0 D. 36

15. HW#5 – Page 1 1. f(4) = 4 2. g(2) = -4 3. g(-3) = 21 4. f(-5) = -14 5. f(3x) =6x – 4 6. f(g(-2)) = 20 7. g(t) = t 2 – 4t 8. f(h) = 2h – 4 9. f(q + 1) = 2q – 2 10. f(2) + g(-2) = 12 11. g(-b) = b 2 + 4b 12. f(r – 1) = 2r – 6

16. HW#5 – Page 1 13.

17. SWBAT… define a function, learn function notation, and a evaluate function Agenda 1. WU ( 5min) 2. Notes on functions with practice problems (45 min) Warm-Up: 1. Write your HW in your planner for the week. HW#5: Functions: Page 2 – 3 Mon, 11/5

18. Objectives Today 1. To learn function mapping 2. To conduct the vertical line test 3. To find the domain and range of a function

19. Function Mapping A set of points or equation where every input has exactly one output. In other words, the domain or x value can not be repeated

20. This is a function! There is only one arrow coming from each x. In other words, x can not be repeated This is a function! There is only one arrow coming from each x There is only one y for each x. It just so happens that it's always the same y for each x. Function Mapping (cont’d)

21. This is not a function. There are two arrows coming from the number 1. The number 1 is associated with two different range elements. In order words, x is repeated. Function Mapping (cont’d)

22. Vertical Line Test No mater where we drop a vertical line, if the vertical line only hits the graph once, it is a function. (Note: I didn’t say anything about the x-axis) So, this graph is a function! What is this function called?  Quadratic function (2, 4) (-2, 4)

23. y x Q: Is the graph a function? Explain. A: Yes, this graph is a function because it passes the vertical line test; wherever you drop a vertical line, it will only hit the graph once. Q: What is the graph called? A: A line Q: What is the function called? A: A linear function Write the equation of the line in slope- intercept form, y = mx + b. Q: What is the slope, m? A: 3 Q: What is the y-intercept, b? A: -2 Vertical Line Test (cont’d) y = 3x – 2

24. Draw a graph, that would NOT pass the vertical line test.

25. Vertical Line Test (cont’d) Intersect at two points These graphs are not functions

26. Domain and Range Domain: What can go into a function. The set of all x values in a function. How “wide” the function is. Range: What comes out of a function. The set of all y values in a function. How “tall” the function is.

27. What is the domain and range of f(x) = x + 1? Domain: All real numbers because there are no restrictions on the domain. Range: All real numbers because there are no restrictions on the range.

28. Domain and Range (cont’d) f(x) = x 2 – 2 The domain is all real numbers. Explanation: There are no restrictions on the domain, the x value. The range is y ≥ -2. Explanation: The graph will never be below -2. (2, 4) (-2, 4) (0, -2)

29. x f(x) =1/2x 2 -4f(-4) = 8 -2f(-2) = 2 f(-1) = 0.5 0f(0) = 0 1f(1) = 0.5 2f(2) = 2 4f(4) = 8 The domain is all real numbers. Explanation: There are no restrictions on the domain, the x value. The range is y ≥ 0. Explanation: The graph will never be below 0.

30. Domain and Range (cont’d) Question: For the function f(x) = x 2, if the domain is {1, 2, 3}, what is the range?

31. Domain and Range (cont’d) Example: {(2, 4), (4, 5), (7, 3)} Question: What is the domain? What is the range? Is this a function? Explain. Example: {(2, 4), (2, 5), (7, 3)} Questions: What is the domain? What is the range? Is this a function? Explain.

32. Revisit our objectives Today: 1. To learn function mapping 2. To conduct the vertical line test 3. To find the domain and range of a function WE MET OUR OBJECTIVES TODAY!!!!