One-to-one and Inverse Functions Digital Lesson One-to-one and Inverse Functions

Definition: Functions A function y = f (x) with domain D is one-to-one on D if and only if for every x1 and x2 in D, f (x1) = f (x2) implies that x1 = x2. 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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Functions

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

Example: Horizontal Line Test Example: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one. a) y = x3 b) y = x3 + 3x2 – x – 1 x y -4 4 8 x y -4 4 8 one-to-one not one-to-one Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Horizontal Line Test

Inverse relation x = |y| + 1 Every function y = f (x) has an inverse relation x = f (y). Domain Range 2 1 0 -1 -2 x y 3 2 1 Function y = |x| + 1 Domain Range Inverse relation x = |y| + 1 2 1 0 -1 -2 x y 3 2 1 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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Inverse Relation

The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}. 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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Ordered Pairs

Example: Ordered Pairs The graphs of a relation and its inverse are reflections in the line y = x. Example: Find the graph of the inverse relation geometrically from the graph of f (x) = x y 2 -2 The ordered pairs of f are given by the equation . y = x The ordered pairs of the inverse are given by . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Ordered Pairs

Example: Inverse Relation Algebraically To find the inverse of a relation algebraically, interchange x and y and solve for y. 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. y = Solve for y. To calculate a value for the inverse of f, subtract 2, then divide by 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Inverse Relation Algebraically

The inverse function of y = f (x) is written y = f -1(x). 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). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Inverse Function

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

Composition of Functions 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) = x3 is f -1(x) = . 3 f -1( f(x)) = = x and f ( f -1(x)) = ( )3 = x. 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Composition of Functions

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