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4-1: Relations and Functions Objectives: • Understand, draw, and determine if a relation is a function. • Graph & write linear equations, determine domain and range. Relations & Functions Relation: a set of ordered pairs Domain: the set of x-coordinates Range: the set of y-coordinates When writing the domain and range, do not repeat values. Relations and Functions Given the relation: {(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)} State the domain: D: State the range: R: Relations and Functions • Relations can be written in several ways: ordered pairs, tables, graphs, or mapping. • We have already seen relations represented as ordered pairs. Table {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} x 3 7 0 -2 -5 3 y 4 2 -1 2 0 3 Mapping • Create two ovals with the domain on the left and the range on the right. • Elements are not repeated. • Connect elements of the domain with the corresponding elements in the range by drawing an arrow. Mapping {(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)} Functions • A function is a relation in which the members of the domain (x-values) DO NOT repeat. • So, for every x-value there is only one y-value that corresponds to it. • y-values can be repeated. Do the ordered pairs represent a function? Ex. 1 {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} Ex. 2 {(4, 1), (5, 2), (8, 2), (9, 8)} Graphs of a Function Vertical Line Test: If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function. Does the graph represent a function? Name the domain and range. x Yes or No D: R: x Yes or No D: R: y y Does the graph represent a function? Name the domain and range. x Yes or No D: R: x Yes or No D: R: y y Does the graph represent a function? Name the domain and range. x Yes or No D: R: x Yes or No D: R: y y Function Notation • When we know that a relation is a function, the “y” in the equation can be replaced with f(x). • f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’. • The ‘f’ names the function, the ‘x’ tells the variable that is being used. Value of a Function Since the equation y = x - 2 represents a function, we can also write it as f(x) = x - 2. Find f(4): Value of a Function If g(s) = 2s + 3, find g(-2). Value of a Function If h(x) = 4x2 – 3x + 1, find h(-2) Find the Domain & Range when given the Equation 1.) f(x) = 1/4x – 5 2.) f(x) = (x – 3)2 3.) f(x) = |x| - 4 4.) f(x) = -x2 + 3