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How can you determine if you have a function? • Vertical Line Test • NO Duplicated “x” values Graphing Inverse Relations Section 6-4 Pages 438-443 Objectives • I can graph functions and their inverses • I can use the Horizontal Line Test Inverse Concept • The main concept of an inverse is the x and y coordinates have switched places ( x, y ) ( y, x) Inverses • The inverse of any relation is obtained by switching the coordinates in each ordered pair of the relation. • Example: • { (1, 2), (3, -1), (5, 4)} is a relation • { (2, 1), (-1, 3), (4, 5) is the inverse. The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation. Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}. The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}. The domain of the inverse relation is the range of the original function. The range of the inverse relation is the domain of the original function. GRAPHING AN INVERSE FUNCTION Given the graph of a relation, the graph of its inverse is obtained by switching x- and ycoordinates. The resulting graph is reflected about the line y = x. 7 Graphing an Inverse • Pick some Critical Points off Original Graph (x, y) • SWITCH the x and y values • Re-plot the newly formed ordered pairs. 9 GRAPH the inverse The graphs of a relation and its inverse are reflections in the line y = x. Example: Find the graph of the inverse relation 3 ( x 2) geometrically from the graph of f (x) = 4 y The ordered pairs of f are 3 ( x 2) given by the equation y . 4 The ordered pairs of the inverse are 3 ( y 2) . given by x 4 y=x 2 x -2 ( y 2) x 4 3 2 -2 ( x 3 2) y 4 Horizontal Line Test A function y = f(x) is one-to-one if and only if no horizontal line intersects the graph of y = f(x) in more than one point. Example: The function y = x2 – 4x + 7 is not one-to-one on the real numbers because the line y = 7 intersects the graph at both (0, 7) and (4, 7). y (4, 7) (0, 7) y=7 2 x 2 Example: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one. b) y = x3 + 3x2 – x – 1 a) y = x3 y -4 y 8 8 4 4 4 -4 4 x one-to-one x not one-to-one Home Work • Worksheet 10-1