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One-to-one and Inverse Functions Digital Lesson

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 A function y = f (x) with domain D is one-to-one on D if and only if for every x 1 and x 2 in D, f (x 1 ) = f (x 2 ) implies that x 1 = x 2. A function is a mapping from its domain to its range so that each element, x, of the domain is mapped to one, and only one, element, f (x), of the range. A function is one-to-one if each element f (x) of the range is mapped from one, and only one, element, x, of the domain. Definition: Functions

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 x y 2 2 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. y = 7 Example: The function y = x 2 – 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). (0, 7) (4, 7) Horizontal Line Test

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 one-to-one Example: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one. a) y = x 3 b) y = x 3 + 3x 2 – x – 1 not one-to-one x y x y Example: Horizontal Line Test

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 DomainRange Inverse relation x = |y| x y Domain Range x y Function y = |x| + 1 Every function y = f (x) has an inverse relation x = f (y). The ordered pairs of : y = |x| + 1 are {(-2, 3), (-1, 2), (0, 1), (1, 2), (2, 3)}. x = |y| + 1 are {(3, -2), (2, -1), (1, 0), (2, 1), (3, 2)}. The inverse relation is not a function. It pairs 2 to both -1 and +1. Inverse Relation

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 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. Ordered Pairs

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 y = x The graphs of a relation and its inverse are reflections in the line y = x. The ordered pairs of f are given by the equation. Example: Find the graph of the inverse relation geometrically from the graph of f (x) = x y The ordered pairs of the inverse are given by. Example: Ordered Pairs

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Find the inverse relation algebraically for the function f (x) = 3x + 2. y = 3x + 2 Original equation defining f x = 3y + 2 Switch x and y. 3y + 2 = x Reverse sides of the equation. To calculate a value for the inverse of f, subtract 2, then divide by 3. y = Solve for y. To find the inverse of a relation algebraically, interchange x and y and solve for y. Example: Inverse Relation Algebraically

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 For a function y = f (x), the inverse relation of f is a function if and only if f is one-to-one. For a function y = f (x), the inverse relation of f is a function if and only if the graph of f passes the horizontal line test. If f is one-to-one, the inverse relation of f is a function called the inverse function of f. The inverse function of y = f (x) is written y = f -1 (x). Inverse Function

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 y = f(x) y = x y = f -1 (x) Example: From the graph of the function y = f (x), determine if the inverse relation is a function and, if it is, sketch its graph. The graph of f passes the horizontal line test. The inverse relation is a function. Reflect the graph of f in the line y = x to produce the graph of f -1. x y Example: Determine Inverse Function

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 The inverse function is an “inverse” with respect to the operation of composition of functions. The inverse function “undoes” the function, that is, f -1 ( f (x)) = x. The function is the inverse of its inverse function, that is, f ( f -1 (x)) = x. Example: The inverse of f (x) = x 3 is f -1 (x) =. 3 f -1 ( f(x)) = = x and f ( f -1 (x)) = ( ) 3 = x Composition of Functions

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 It follows that g = f -1. Example: Verify that the function g(x) = is the inverse of f(x) = 2x – 1. Example: Composition of Functions f(g(x)) = 2g(x) – 1 = 2( ) – 1 = (x + 1) – 1 = x g( f(x)) = = = = x

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