4.6 Formalizing Relations and Functions: Relation: the pairing of numbers in one set mainly the domain with exactly one number of the range represented by ordered pairs (x, y) Domain: The x value of the ordered pair (x, y) Range: The y value of the ordered pair (x, y)

Vertical Line Test: If any vertical line passes through a graph more than once it is a relations but not a function. Not a Function: Hit the graph more than once Function: Only hit the graph once

Graphs:

In math we use a few ways to identify a function: 1. Mapping Diagrams: The use of oval shapes to connect a number of the domain with the number of the range. EX: Identify the domain and range using a mapping diagram then decide if the relation is a function? A: {(-2, 0.5), (0, 2.5), (4, 6.5), (5, 2.5)} B: {(6, 5), (4, 3), (6, 4), (5, 8)}

The relation is a FUNCTION since there is only one y for each x. Domain(x): {-2, 0, 4, 5} Range(y): {0.5, 2.5, 6.5} -2 0.5 2.5 4 6.5 5 The relation is a FUNCTION since there is only one y for each x.

B: {(6, 5), (4, 3), (6, 4), (5, 8)} Domain(x):{4, 5, 6} Range(y): {3, 4, 5, 8} 3 4 4 5 5 6 8 The relation is a NOT A FUNCTION since the six (x) has two (y) values: 4 and 5.

YOU TRY IT: EX: Identify the domain and range using a mapping then decide if the relation a function? A: {(4.2, 1.5), (5, 2.2), (7, 4.8), (4.2, 0)} B: {(-1, 1), (-2, 2), (4, -4), (7, -7)}

The relation is a NOT A FUNCTION since there is two y’s for 4.2 . Domain(x): {4.2, 5, 7} Range(y): {0, 1.5, 2.2, 4.8} 4.2 5 1.5 7 2.2 4.8 The relation is a NOT A FUNCTION since there is two y’s for 4.2 .

The relation is a FUNCTION since there is exactly one y for each x . Domain(x): {-2, -1, 4, 7} Range(y): {-7, -4, 1, 2} -2 -7 -1 -4 4 1 7 2 The relation is a FUNCTION since there is exactly one y for each x .

IDENTIFYING FUNCTIONS: A relation is a function if it passes the vertical line test. Decide if the relation is a function. Provide domain and Range of the graph A B C D E

A relation is a function if it passes the vertical line test. A: Relation that is a function B: Relation that is Not a function C: Relation not Function

A relation is a function if it passes the vertical line test. D: Relation that is a function E: Relation that is Not a function since 3 has two y values.

DOMAIN : x values of our graph: RANGE: y values of our graph D: { x | -∞< x < ∞} R: { y |0 < y < ∞} D: { x | 0 < x < ∞} R: { y | - ∞ < y < ∞} D: { x | -7 < x < 7} R: { y |-2 < y < 2}

DOMAIN : x values of our graph: RANGE: y values of our graph D: { x | -∞< x < ∞} R: { y | - ∞ < y < ∞} D: { 1, 2, 3} R: { a, b, c, d}

EVALUATION FUNTIONS: We evaluate a function by plugging in a given value of x. Evaluate f(4) A B A person can type at a speed of w(x) = 250x. How many words will it be in 8 minutes? E C

EVALUATION FUNTIONS: We evaluate a function by plugging in a given value of x. Evaluate f(4) Looking at the value of x = 4, Looking at x = 4, we see that the values of y are: - 2, and + 2 we see that the values of y are about: - 1.8, and + 1.8

EVALUATION FUNTIONS: We evaluate a function by plugging in a given value of x. Evaluate f(4) A person can type at a speed of W(x) = 250x. How many words will it be in 8 minutes? Y = x3 We need to replace x for 8. W(x) =250x W(8) = 250(8) W(8) = 2000 words/minute Looking at the value of x = 4, we see that the value of y will be 64, not in the graph.

Provide the domain and range of the following: YOU TRY IT: Provide the domain and range of the following:

Domain: {x | 𝝅 𝟔 < x < 𝟓𝝅 𝟔 } YOU TRY IT (Solution): Domain: {x | 𝝅 𝟔 < x < 𝟓𝝅 𝟔 } RANGE: { y | -2 < y < 2}

CLASSWORK: Page 271-273 Problems: As many as needed to master the concept.