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Do Now Complete the chart for each linear equation. y=x-2 x y = 2x + 3 y x 4 2 -3 -2 0 0 2 1 -1 -1 y Advanced Algebra Trigonometry Appendix C Functions Objective: Determine if a relation is a function, and find the domain and range. Relations Relation: a set of ordered pairs. { ( -3, 2), (-1, 1 ), ( 0, 7), (2, 4), (4, 3)} {( -2, 1), (-1, 2), ( 0, 3), (1, 4), (2, 5)} Independent Variable ( x, y ) Dependent Variable The value of y “depends” on the value of x. Domain: the set of all x-coordinates, independent variable Range: the set of all y-coordinates, dependent variable Relations 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} Functions Function: a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component. • 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. Ways to Represent a Function • Symbolic x,y y 2x or y 2x • Numeric • Graphical X Y 1 2 5 10 -1 -2 3 6 • Verbal The cost is twice the original amount. Does the relation represent a function? H = {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} No, 3 is repeated in the domain. G = {(4, 1), (5, 2), (8, 2), (9, 8)} Yes, no x-coordinate is repeated. Finding Domain & Range Give the domain & range of each relation. Is it a function? Example 1 Example 2 x y 1 1 -1 1 5 1 3 1 {(3, -6), (1, 3), (-2, 4), (0,3), (1, -2), (3, 0)} Vertical Line Test 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. Finding Domain & Range Give the domain & range of each relation. Is it a function? Finding Domain & Range Give the domain & range of each relation. Is it a function? ... . y x Does the graph represent a function? Name the domain and range. x Function: Yes D: All real numbers R: All real numbers y x y Function: Yes D: All real numbers R: y ≥ -6 Does the graph represent a function? Name the domain and range. x Function: No D: x ≥ 1/2 R: All real numbers y x y Function: No D: All real numbers R: All real numbers 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. • The parenthesis DO NOT mean multiplication! • f(x) is another name for y. • Sometimes other letters such as g, h or capital letters F, G and H are used to name functions. Using Function Notation Find the value of each function. 1. If g(s) = 2s + 3, find g(-2). 2. If h(x) = x2 - x + 7, find h(2). 3. If f(x) = -x2 + 5x – 3 find f(q) Homework Page 457-458 #’s 12, 13, 18, 19, 35, 36, 37, 46