WarmUp for 101413 You have been given several

Warm-Up for 10/14/13: You have been given several relations. You should decide which relations you think qualify as “functions” and which ones do not. Sort them with your partner on the placemat. You have 5 minutes. 5: 00 4: 59 4: 58 4: 57 4: 56 4: 48 4: 51 4: 53 4: 47 4: 46 4: 45 4: 52 4: 54 4: 50 4: 49 4: 44 4: 43 2: 41 1: 03 4: 11 3: 11 2: 11 1: 11 0: 11 4: 42 4: 41 4: 40 4: 39 4: 38 4: 37 4: 36 4: 35 4: 34 4: 33 4: 32 4: 31 4: 30 4: 29 4: 28 4: 27 4: 26 4: 25 4: 24 4: 23 4: 22 4: 21 4: 20 4: 19 4: 18 4: 17 4: 16 4: 15 4: 14 4: 13 4: 12 4: 10 4: 09 4: 08 4: 07 4: 06 4: 05 4: 04 4: 03 4: 02 4: 01 4: 00 3: 59 3: 58 3: 57 3: 56 3: 55 3: 54 3: 53 3: 52 3: 51 3: 50 3: 49 3: 48 3: 47 3: 46 3: 45 3: 44 3: 43 3: 42 3: 41 3: 40 3: 39 3: 38 3: 37 3: 36 3: 35 3: 34 3: 33 3: 32 3: 31 3: 30 3: 29 3: 28 3: 27 3: 26 3: 25 3: 24 3: 23 3: 22 3: 21 3: 20 3: 19 3: 18 3: 17 3: 16 3: 15 3: 14 3: 13 3: 12 3: 10 3: 09 3: 08 3: 07 3: 06 3: 05 3: 04 3: 03 3: 02 3: 01 3: 00 2: 59 2: 58 2: 57 2: 56 2: 55 2: 54 2: 53 2: 52 2: 51 2: 50 2: 49 2: 48 2: 47 2: 46 2: 45 2: 44 2: 43 2: 42 2: 40 2: 39 2: 38 2: 37 2: 36 2: 35 2: 34 2: 33 2: 32 2: 31 2: 30 2: 29 2: 28 2: 27 2: 26 2: 25 2: 24 2: 23 2: 22 2: 21 2: 20 2: 19 2: 18 2: 17 2: 16 2: 15 2: 14 2: 13 2: 12 2: 10 2: 09 2: 08 2: 07 2: 06 2: 05 2: 04 2: 03 2: 02 2: 01 2: 00 1: 59 1: 58 1: 57 1: 56 1: 55 1: 54 1: 53 1: 52 1: 51 1: 50 1: 49 1: 48 1: 47 1: 46 1: 45 1: 44 1: 43 1: 42 1: 41 1: 40 1: 39 1: 38 1: 37 1: 36 1: 35 1: 34 1: 33 1: 32 1: 31 1: 30 1: 29 1: 28 1: 27 1: 26 1: 25 1: 24 1: 23 1: 22 1: 21 1: 20 1: 19 1: 18 1: 17 1: 16 1: 15 1: 14 1: 13 1: 12 1: 10 1: 09 1: 08 1: 07 1: 06 1: 05 1: 04 1: 02 1: 01 1: 00 0: 59 0: 58 0: 57 0: 56 0: 55 0: 54 0: 53 0: 52 0: 51 0: 50 0: 49 0: 48 0: 47 0: 46 0: 45 0: 44 0: 43 0: 42 0: 41 0: 40 0: 39 0: 38 0: 37 0: 36 0: 35 0: 34 0: 33 0: 32 0: 31 0: 30 0: 29 0: 28 0: 27 0: 26 0: 25 0: 24 0: 23 0: 22 0: 21 0: 20 0: 19 0: 18 0: 17 0: 16 0: 15 0: 14 0: 13 0: 12 0: 10 0: 09 0: 08 0: 07 0: 06 0: 05 0: 04 0: 03 0: 02 0: 01 0: 00 4: 49 4: 44 4: 43 2: 41 1: 03 0: 11 4: 11 3: 11 2: 11 1: 11 4: 42 4: 41 4: 40 4: 39 4: 38 4: 37 4: 36 4: 35 4: 34 4: 33 4: 32 4: 31 4: 30 4: 29 4: 28 4: 27 4: 26 4: 25 4: 24 4: 23 4: 22 4: 21 4: 20 4: 19 4: 18 4: 17 4: 16 4: 15 4: 14 4: 13 4: 12 4: 10 4: 09 4: 08 4: 07 4: 06 4: 05 4: 04 4: 03 4: 02 4: 01 4: 00 3: 59 3: 58 3: 57 3: 56 3: 55 3: 54 3: 53 3: 52 3: 51 3: 50 3: 49 3: 48 3: 47 3: 46 3: 45 3: 44 3: 43 3: 42 3: 41 3: 40 3: 39 3: 38 3: 37 3: 36 3: 35 3: 34 3: 33 3: 32 3: 31 3: 30 3: 29 3: 28 3: 27 3: 26 3: 25 3: 24 3: 23 3: 22 3: 21 3: 20 3: 19 3: 18 3: 17 3: 16 3: 15 3: 14 3: 13 3: 12 3: 10 3: 09 3: 08 3: 07 3: 06 3: 05 3: 04 3: 03 3: 02 3: 01 3: 00 2: 59 2: 58 2: 57 2: 56 2: 55 2: 54 2: 53 2: 52 2: 51 2: 50 2: 49 2: 48 2: 47 2: 46 2: 45 2: 44 2: 43 2: 42 2: 40 2: 39 2: 38 2: 37 2: 36 2: 35 2: 34 2: 33 2: 32 2: 31 2: 30 2: 29 2: 28 2: 27 2: 26 2: 25 2: 24 2: 23 2: 22 2: 21 2: 20 2: 19 2: 18 2: 17 2: 16 2: 15 2: 14 2: 13 2: 12 2: 10 2: 09 2: 08 2: 07 2: 06 2: 05 2: 04 2: 03 2: 02 2: 01 2: 00 1: 59 1: 58 1: 57 1: 56 1: 55 1: 54 1: 53 1: 52 1: 51 1: 50 1: 49 1: 48 1: 47 1: 46 1: 45 1: 44 1: 43 1: 42 1: 41 1: 40 1: 39 1: 38 1: 37 1: 36 1: 35 1: 34 1: 33 1: 32 1: 31 1: 30 1: 29 1: 28 1: 27 1: 26 1: 25 1: 24 1: 23 1: 22 1: 21 1: 20 1: 19 1: 18 1: 17 1: 16 1: 15 1: 14 1: 13 1: 12 1: 10 1: 09 1: 08 1: 07 1: 06 1: 05 1: 04 1: 02 1: 01 1: 00 0: 59 0: 58 0: 57 0: 56 0: 55 0: 54 0: 53 0: 52 0: 51 0: 50 0: 49 0: 48 0: 47 0: 46 0: 45 0: 44 0: 43 0: 42 0: 41 0: 40 0: 39 0: 38 0: 37 0: 36 0: 35 0: 34 0: 33 0: 32 0: 31 0: 30 0: 29 0: 28 0: 27 0: 26 0: 25 0: 24 0: 23 0: 22 0: 21 0: 20 0: 19 0: 18 0: 17 0: 16 0: 15 0: 14 0: 13 0: 12 0: 10 0: 09 0: 08 0: 07 0: 06 0: 05 0: 04 0: 03 0: 02 0: 01 0: 00

Relations and Functions UNIT QUESTION: How can we use realworld situations to construct and compare linear and exponential models and solve problems? Standards: MCC 9 -12. A. REI. 10, 11, F. IF. 1 -7, 9, F. BF. 1 -3, F. LE. 1 -3, 5 Today’s Question: What is a function, and how is function notation used to evaluate functions? Standard: MCC 9 -12. F. IF. 1 and 2

• A relation is any set of ordered pairs. • The domain is the set of all the first elements of the ordered pairs. (all of the x-values) • The range is the set of all the second elements of the ordered pairs. (all of the yvalues)

Relations can be represented in different ways. Ordered pairs Table (3, 3) (– 1, 2) (0, – 2) Mapping -1 -2 0 2 3 3 x y -1 2 0 -2 3 3 Graph

State the domain and range of each relation. 1. {(1, 3), (2, 4), (3, 3), (4, 4)} Domain: {1, 2, 3, 4 } Range: {3, 4 } {(-1, 2), (2, 4), (3, 6)} 2. -1 2 2 4 Domain: {-1, 2, 3 } 3 6 Range: {2, 4, 6}

3. Domain: {1, 2, 3} Range: {1, 2, 3} {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}

Discrete Graph – a graph whose points _____ are not connected. are Continuous Graph – a graph whose points _____ connected. ● ● ● Continuous _____ graph Discrete _____ graph

State the domain and range of the continuous graphs: 4. D= [-2, 2] R= [-1, 4]

5. D: (-∞, ∞) or “All Real Numbers” R: [-2, ∞)

Function: a relation in which each element in the domain is paired with exactly one element in the range. S = {(0, 3), (-2, 5), (4, 7)} ______a function! is Each x is paired with exactly one y! is not T = {(1, -2), (2, 3), (1, 4)} _______a function! The element 1 is paired with -1 and 4!

Consider each mapping. Is the relation a function? x 1 8 6 y x 3 2 Not a function The element 8 is paired with 3 and 2 4 6 -9 10 y -8 6 1 Is a function

Consider a relation represented in tabular form. Which relation is a function? (a) x -2 3 4 -2 y 0 4 -1 9 Not a function The element -2 is paired with 0 and 9! (b) x -1 2 5 -3 y 4 5 Is a function

Consider a relation that is represented in graphical form: (discrete graph below!) Suggestion: Write as a set of ordered pairs. T = {(1, -1), (2, 3), (1, 4)} Not a function: The element 1 is paired with -1 and 4.

For relations represented as graphs, try this test: FAIL! Vertical Line Test: If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.

Determine whether each relation is a function: (a) PASS! Is a function y (b) y FAIL! x Not a function (c) PASS! x Is a function

Identify each graph as continuous or discrete: Discrete (a) y y (b) (c) x x Continuous

Equations: Function Notation: y = 3 x + 1 f(x) = 3 x + 1 y = 4 – 2 x g(x) = 4 – 2 x y = x 2 + 3 x – 6 h(x) = x 2 + 3 x – 6 The expression f(x) is read “f of x” and represents the variable y. f(3) represents the y value for the function, f, when x = 3.

Closing for 10/14/13: Go back to the Function Sorting Activity from the Warm-Up. Discuss with your partner if you need to make any changes to your original decisions on what was a function, and why. Then we will share. 5: 00 4: 59 4: 58 4: 57 4: 56 4: 48 4: 51 4: 53 4: 47 4: 46 4: 45 4: 52 4: 54 4: 50 4: 49 4: 44 4: 43 2: 41 1: 03 4: 11 3: 11 2: 11 1: 11 0: 11 4: 42 4: 41 4: 40 4: 39 4: 38 4: 37 4: 36 4: 35 4: 34 4: 33 4: 32 4: 31 4: 30 4: 29 4: 28 4: 27 4: 26 4: 25 4: 24 4: 23 4: 22 4: 21 4: 20 4: 19 4: 18 4: 17 4: 16 4: 15 4: 14 4: 13 4: 12 4: 10 4: 09 4: 08 4: 07 4: 06 4: 05 4: 04 4: 03 4: 02 4: 01 4: 00 3: 59 3: 58 3: 57 3: 56 3: 55 3: 54 3: 53 3: 52 3: 51 3: 50 3: 49 3: 48 3: 47 3: 46 3: 45 3: 44 3: 43 3: 42 3: 41 3: 40 3: 39 3: 38 3: 37 3: 36 3: 35 3: 34 3: 33 3: 32 3: 31 3: 30 3: 29 3: 28 3: 27 3: 26 3: 25 3: 24 3: 23 3: 22 3: 21 3: 20 3: 19 3: 18 3: 17 3: 16 3: 15 3: 14 3: 13 3: 12 3: 10 3: 09 3: 08 3: 07 3: 06 3: 05 3: 04 3: 03 3: 02 3: 01 3: 00 2: 59 2: 58 2: 57 2: 56 2: 55 2: 54 2: 53 2: 52 2: 51 2: 50 2: 49 2: 48 2: 47 2: 46 2: 45 2: 44 2: 43 2: 42 2: 40 2: 39 2: 38 2: 37 2: 36 2: 35 2: 34 2: 33 2: 32 2: 31 2: 30 2: 29 2: 28 2: 27 2: 26 2: 25 2: 24 2: 23 2: 22 2: 21 2: 20 2: 19 2: 18 2: 17 2: 16 2: 15 2: 14 2: 13 2: 12 2: 10 2: 09 2: 08 2: 07 2: 06 2: 05 2: 04 2: 03 2: 02 2: 01 2: 00 1: 59 1: 58 1: 57 1: 56 1: 55 1: 54 1: 53 1: 52 1: 51 1: 50 1: 49 1: 48 1: 47 1: 46 1: 45 1: 44 1: 43 1: 42 1: 41 1: 40 1: 39 1: 38 1: 37 1: 36 1: 35 1: 34 1: 33 1: 32 1: 31 1: 30 1: 29 1: 28 1: 27 1: 26 1: 25 1: 24 1: 23 1: 22 1: 21 1: 20 1: 19 1: 18 1: 17 1: 16 1: 15 1: 14 1: 13 1: 12 1: 10 1: 09 1: 08 1: 07 1: 06 1: 05 1: 04 1: 02 1: 01 1: 00 0: 59 0: 58 0: 57 0: 56 0: 55 0: 54 0: 53 0: 52 0: 51 0: 50 0: 49 0: 48 0: 47 0: 46 0: 45 0: 44 0: 43 0: 42 0: 41 0: 40 0: 39 0: 38 0: 37 0: 36 0: 35 0: 34 0: 33 0: 32 0: 31 0: 30 0: 29 0: 28 0: 27 0: 26 0: 25 0: 24 0: 23 0: 22 0: 21 0: 20 0: 19 0: 18 0: 17 0: 16 0: 15 0: 14 0: 13 0: 12 0: 10 0: 09 0: 08 0: 07 0: 06 0: 05 0: 04 0: 03 0: 02 0: 01 0: 00

Let f(x) = 3 x +1 and h(x) = x 2 + 3 x – 6. Find the value of each of the following: (1). f(-2) = 3(-2) + 1 = -6 + 1 = -5 For the graph of f, when x = -2, y = -5. (2). h(-3) = (-3)2 + 3(-3) – 6 =9– 9– 6 = -6 For the graph of h, when x = -3, y = -6.

Find the value of each expression given its graph. 3 (1). f(4) = _____ Discrete graph (2). g(-1) = ______ -2 Continuous graph

Find the value of each expression given its graph. 2 (3). f(-2) = _____ D = {-2, -1, 1, 3, 4} R = { -1, 0, 2, 3} (4). g(-3) = ______ -4 D = {x| x ≥ -5} R = { y| y ≥ -4}

For the equation y = 3 x + 2 we say that • y is written as a function of x • y is written in terms of x. x is called the independent variable. y is called the dependent variable.

For the equation A = πr 2 we say that • A is written as a function of r • A is written in terms of r. r is called the independent variable. A is called the dependent variable.