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1-6 and 1- 7: 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: {0,1, 2, 3} State the range: R: {-6, 0, 4} Relations and Functions • Another way to represent a relation is to use a mapping diagram. • 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)} 2 1 0 3 -6 4 0 Mapping The elements of the domain are called inputs, and the elements of the range are called outputs. Relations and Functions • Sometimes a relation is represented by a table. x y {(3, 4), (7, 2), (0, -1), 3 4 7 2 (-2, 2), (-5, 0), (3, 3)} 0 -2 -5 3 -1 2 0 3 Relations and Functions • Another way to represent a relation is to draw its graph. {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} 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. Functions • When no two elements of the domain are mapped to the same elements of the range, it is called one-to-one mapping. Do the ordered pairs represent a function? {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} No, 3 is repeated in the domain. {(4, 1), (5, 2), (8, 2), (9, 8)} Yes, no x-coordinate is repeated. Graphs of a Function A horizontal line is a constant function. A linear function is a function whose graph is a line or part of a line. Graphs of a Function If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function. Is the graph a function? Why/not? x y x y Is the graph a function? Why/not? x y x 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): f(4) = 4 - 2 f(4) = 2 Value of a Function If g(s) = 2s + 3, find g(-2). g(-2) = 2(-2) + 3 =-4 + 3 = -1 g(-2) = -1