Inverse Functions

One-to-one and Inverse Functions Review: A is any set of ordered pairs. A is a set of ordered pairs where x is not repeated. A repeated. function does not have any y values Only functions. functions can have 2 What is an Inverse? An inverse relation is a relation that performs the opposite operation on x (the domain). Examples: f(x) = x – 3 g(x) = x , x ≥ 0 h(x) = 2x k(x) = -x + 3 f-1(x) = x + 3 g-1(x) = x2 , x ≥ 0 h-1(x) = ½ x k-1(x)= -(x – 3) 3 Section 1.9 : Illustration of the Definition of Inverse Functions 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 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. 5 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. 6 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. 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). y (4, 7) (0, 7) y=7 2 x 2 7 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 x one-to-one The Inverse is a Function 4 x not one-to-one The Inverse is a Relation 8 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) = 1 ( x 3 − 2) 4 y The ordered pairs of f are given by the equation y = 1 ( x 3 − 2) . y=x 2 4 The ordered pairs of the inverse are given by x = 1 ( y 3 − 2) . x -2 4 ( y − 2) x= 4 3 2 -2 ( x 3 − 2) y= 4 9 Section 1.9 : Figure 1.93, Graph of an Inverse Function Functions and their inverses are symmetric over the line y =x 10 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. 11 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) = 3 x . f -1( 3 3 f(x)) = x = x and f ( f -1(x)) = ( 3 x )3 = x. 13 x +1 Example: Verify that the function g(x) = is the inverse of f(x) = 2x – 1. 2 ( f ( x) + 1) (( 2 x − 1) + 1) 2x g( f(x)) = = = =x 2 2 2 f(g(x)) = 2g(x) – 1 = 2( x + 1 ) – 1 = (x + 1) – 1 = x 2 It follows that g = f -1. 14 Now Try: Page 99 #13, 15, 23 pg 101 # 69, 71, 73 15 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 • Original and Inverses are symmetric over y =x • “ “ ” have reverse domain & ranges • Given two relations to test for inverses. f(f-1(x)) = x and f-1(f(x)) = x **both must be true** 16 Practice Problems and Homework • Page 99-100 # 16, 18, 20, 24, 39, 41, 43 #55-67 odd 17