Drill #16 List the relation (set of ordered pairs) and the domain and range of the following mapping: 1. Graph the following relation, state the domain and identify if it a function 2. {(3,-4), (-2, 4), (-3,4)} 3. {(1,-2), (-1, 2), (-1,4)} y x 2 3 -1 4

2-1 Relations and Functions Objective: To graph a relation, state its domain and range, and determine if it is a function, and to find values of functions for given elements in a domain.

Cartesian Coordinate Plane Cartesian Coordinate Plane: Composed of an x-axis (horizontal) and y-axis (vertical) which meet at the origin and divide the plane into four quadrants. x – axis: The horizontal axis in the coordinate plane. y – axis: The vertical axis in the coordinate plane. origin: The point where the x-axis meets the y-axis corresponding the coordinate (0,0)

The Coordinate Plane x y Quadrant II ( - , + ) Quadrant I ( + , + ) (0,0) Origin Quadrant IV ( + , - ) Quadrant III ( - , - ) y

Relation, Domain, and Range relation: A set of ordered pairs. domain: The set of all the x – coordinates (the 1st numbers) of a relation. For a function, it’s the set of all possible values of x. range: The set of all the y – coordinates (the 2nd numbers) of a relation. For a function, it’s the set of all possible values of y. Example: Name the domain and range of the following relation: { (-1, 2), (-1, 3), (1, 3) }

Mapping mapping: Shows how each element of the domain is paired with each element of the range. Example: { (-1, 2), (-1, 3), (-1, 4) } D R 2 -1 3 4

Functions function: A special type of relation in which each element of the domain is paired with exactly one element of the range. (no x- values are repeated) NOTE : In a function, every x – value (input) has exactly one y – value (output). discrete function: a function that consists of points that are not connected.

Continuous Functions x y continuous functions: A function that can be graphed with a line or a smooth curve and has a domain with an infinite number of elements. 1 x -1 (0,0) Origin y

Vertical Line Test Vertical Line Test: If a vertical line intersects a graph at more than one point then the relation is not a function. Pass a pencil vertically over a graph. If the pencil ever touches more than one point at a time on the graph, then the graph fails the vertical line test, and is not a function. Every x – value must have a unique y –value.

Mapping: Classwork5 Identify the domain and range of each mapping. State whether or not each is a function: A. B. -3 -1 -1 2 3 3 5 1 4 D D R R

Classwork Draw a mapping of the following relations. State the a) Domain, b) Range of each set. {(1, 2), (1, 3), (1, 4)} {(2, 3), (-1, 3), (1, -3)}

One to One* One to One Functions: A function such that each element of the domain is paired with exactly one unique element in the range. One to one One to one Not one to one D R D R D R -1 2 3 4 -1 3 4 -3 2 3 4 -1 3 1 5

Onto* Onto Functions: A function such that each element of the range is paired with exactly one unique element in the domain. Onto Onto Not onto (not a function) D R D R D R -1 2 3 4 -1 3 4 -3 2 4 -1 3 5 1 5

One to One and Onto* One to one and onto: Each element of the domain is paired with a unique range value, and all range values are paired with a domain value. One to one One to one Not one to one and onto not onto Onto D R D R D R -1 2 3 4 -1 3 4 -3 4 2 4 6 -1 3 1

Horizontal Line Test: One to One Test Horizontal Line Test: If a horizontal line intersects a graph at more than one point then the relation is not one to one. Pass a pencil horizontally over a graph. If the pencil ever touches more than one point at a time on the graph, then the graph fails the vertical line test, and is not a function. Every y – value must have a unique x –value.

Evaluating Functions Evaluate the following for the given functions, giving your answer in function notation: Ex1: f(1) Ex5: g(-1) Ex2: f(-½) Ex6: g(¼) Ex3: f(a) Ex7: g(a) Ex4: f(2x+1) Ex8: g(-t)

Evaluating functions Graph the following function determine the a.) Domain and Range, b.) whether the equation is a function, whether is one-to-one, onto, both or neither, c.) whether it is discrete or continuous. x f(x) -3 -2 -1 1 2 3

Evaluating functions Graph the following function determine the a.) Domain and Range, b.) whether the equation is a function, whether is one-to-one, onto, both or neither, c.) whether it is discrete or continuous. x f(x) -3 -2 -1 1 2 3