1.5 Inverse functions Question Is there any ways to express the statement: “x is an angle such that cos x=1” in a short way? In other words, if f is a function that turns x into y, then what is THE THING THAT TURNS Y BACK INTO X? It is called an Inverse function Any function has an inverse? f (x) 2 x f (x) x f (x) 2 2 f :times 2 x 2x f :power 2 x x2 f divided by 2 square root ? x 2 x x x The inverse is g( x ) x 2 The inverse is g( x ) x only true when x 0. No inverse because there is no function that can turn 2 into any number. In this section, you’ll learn What is an inverse function? How to find an inverse function? When does a function has an inverse? How inverse trigonometric functions are defined? Content Definition of inverse functions Find an inverse function Condition for the existence of an inverse Domain, range, and graph of inverse trigonometric functions What is an inverse function? Given f is a function with domain A and range B. The inverse function f 1 (if exists) has domain B and range A and is defined as follows : f 1 y x f ( x ) y for all y in B Properties: f 1 f ( x ) x for all x A f f 1 ( y ) y for all y B This definition says that if f turns x int o y, then f 1 will turn y back int o x. It means that f 1 reverses the action of f . Example: Find f 1 , if exists, given : a) f (0,3),(1,5),(3,9) b) f ( x ) 2 x 3 2x 3 c) f ( x ) x 1 Solution a) The inverse of f simply reverses the ordered pairs, thus f 1 (3,0),(5,1),(9,3) You have to check to ensure that f 1 is a function, i.e. each x has a unique y. 0 3 1 5 9 3 3 5 0 9 3 1 b) Let y 2 x 3, you have to solve this equation to find x in terms of y. y 3 2x y3 x 2 y3 1 So f ( y ) . 2 2x 3 c) Let y , you have to solve this equation to find x in terms of y. x 1 yx y 2 x 3 ( y 2) x 3 y 3 y x 2y Notice that this expression must be defined and ensure that x 1. So, y 2. y3 1 Therefore, f ( y) for all y 2. 2y DIY: Find f , if exists, given: 1 1) f (0,3),(1,5),(3,9),(5,9) A . f 1 (3,0),(5,1),(9,3),(9,5) B . f 1 does not exist. 2) f ( x ) 2 x A . f 1 ( y ) (2 y)2 C . f 1 ( y ) (2 y)2 , y 2 B . f 1 ( y ) y 2, y 0 D . f 1 ( y ) y 2, y 2 Graph of an inverse The graphs of f and f 1 (if exists) are symmetric with respect to the line y=x. The existence of an inverse If f 1 exists, then its graph must satisfies the Vertical line test. It implies that the graph of f must satisfies the HORIZONTAL LINE TEST. The horizontal line test: A function f has an inverse if and only if the graph of f intersects any horizontal line at AT MOST ONE POINT. f f f 1 f has an inverse. f has no inverse. Trigonometric functions Trigonometric functions DO NOT satisfy the Horizontal line test. However, if we restrict the domain of these functions, the inverse will exist. The sine function The inverse sin 1 x sin 1 x y sin y x and 2 y 2 The cosine function The inverse cos1 x cos1 x y cos y x and 0 y The tangent function The inverse tan 1 x tan 1 x y tan y x and 2 y 2 The cotangent function The inverse cot 1 x cot 1 x y cot y x and 0 y Example: Evaluate 2 a. sin 2 1 1 c. tan 3 b. cos 0 1 1 Solution 2 2 a. sin y sin y 2 2 1 b. cos 0 y cos y 0 1 2 y and 0 y so y 1 1 c. tan y tan y 3 3 1 and and 2 y 2 2 2 so y . so y 6 . 4 . 1 DIY : Find the domain of f ( x ) sin 2 5 x 1 . Properties Notice: These properties are all implied from two properties of an inverse. Can you write the similar properties for cosine and cotangent functions? Example: Evaluate a. sin sin 1 0.5 b. sin sin 1 2 c. sin 1 sin 0.5 Solution Applying the properties in the previous slide, we get: because 1 0.5 1. b. sin sin 2 does not exist because 2 is outside [-1,1] a. sin sin 1 0.5 0.5 1 c. sin 1 sin 0.5 =0.5 because 2 0.5 2 DIY: Let f ( x ) sin 2 cos1 x sin 1 x . Find the domain of f and solve f ( x ) 0. Now, you’re able to Find an inverse function Determine if a function has an inverse State domain, range, graph, and do calculation with inverse trigonometric functions