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1.9 Inverse Functions.ppt
1.
One-to-one and Inverse Functions
2.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 2 Review: A is any set of ordered pairs. A function does not have any y values repeated. A is a set of ordered pairs where x is not repeated. Only functions can have functions.
3.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 3 What is an Inverse? Examples: f(x) = x – 3 f-1(x) = x + 3 g(x) = , x ≥ 0 g-1(x) = x2 , x ≥ 0 h(x) = 2x h-1(x) = ½ x k(x) = -x + 3 k-1(x)= -(x – 3) x An inverse relation is a relation that performs the opposite operation on x (the domain).
4.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 4 Section 1.9 : Illustration of the Definition of Inverse Functions
5.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 5 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 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.
6.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 6 How do we know if an inverse function exists? • Inverse functions only exist if the original function is one to one. Otherwise it is an inverse relation and cannot be written as f-1(x). • What does it mean to be one to one? That there are no repeated y values.
7.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 7 x y 2 2 Horizontal Line Test Used to test if a function is one-to one If the line intersection more than once then it is not one to one. Therefore there is not inverse function. y = 7 Example: The function y = x2 – 4x + 7 is not one-to-one because a horizontal line can intersect the graph twice. Examples points: (0, 7) & (4, 7). (0, 7) (4, 7)
8.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 8 one-to-one The Inverse is a Function 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 not one-to-one The Inverse is a Relation x y -4 4 4 8 x y -4 4 4 8
9.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 9 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 . 4 ) 2 ( 3 y x 4 ) 2 ( 3 x y Example: Find the graph of the inverse relation geometrically from the graph of f(x) = ) 2 ( 4 1 3 x x y 2 -2 -2 2 The ordered pairs of the inverse are given by . ) 2 ( 4 1 3 x y ) 2 ( 4 1 3 y x
10.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 10 Section 1.9 : Figure 1.93, Graph of an Inverse Function Functions and their inverses are symmetric over the line y =x
11.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Find the inverse relation algebraically for the function f(x) = 3x + 2. To find the inverse of a relation algebraically, interchange x and y and solve for y.
12.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 13 DETERMINING IF 2 FUNCTIONS ARE INVERSES: The inverse function “undoes” the original 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) = . x 3 f -1( f(x)) = = x and f ( f -1(x)) = ( )3 = x. 3 x 3 x 3
13.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 14 It follows that g = f -1. Example: Verify that the function g(x) = is the inverse of f(x) = 2x – 1. f(g(x)) = 2g(x) – 1 = 2( ) – 1 = (x + 1) – 1 = x 2 1 x 2 1 x g( f(x)) = = = = x 2 ) 1 ) 1 2 (( x 2 2x 2 ) 1 ) ( ( x f
14.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 15 Now Try: Page 99 #13, 15, 23 pg 101 # 69, 71, 73
15.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 16 Review of Today’s Material • A function must be 1-1 (pass the horizontal line test) to have an inverse function (written f-1(x)) otherwise the inverse is a relation (y =) • To find an inverse: 1) Switch x and y 2) Solve for y • Given two relations to test for inverses. f(f-1(x)) = x and f-1(f(x)) = x **both must be true** • Original and Inverses are symmetric over y =x • “ “ ” have reverse domain & ranges
16.
Copyright © by
Houghton Mifflin Company, Inc. All rights reserved. 17 Practice Problems and Homework • Page 99-100 # 16, 18, 20, 24, 39, 41, 43 #55-67 odd
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