5.3 Inverse Functions.notebook January 06, 2017 5.3 Inverse Functions True or False? 1. The inverse of a function is a function. 2. If g is the inverse function of f, then f is the inverse function of g. 3. The domain of f ‐1 is equal to the range of f, and the range of f ‐1 is equal to the domain of f. 4. A function need not have an inverse function, but if it does, the inverse function is unique. 5. f(x) = √x is the inverse of g(x) = x2 . 6. The graph of f contains the point (a, b) if and only if the graph of f ‐1 contains the point (b, a). 7. f has an inverse function if and only if every horizontal line intersects the graph of f at most once. 8. Graphs of inverse functions have reciprocal slopes. Intro/Warm­up 1 5.3 Inverse Functions.notebook January 06, 2017 Jan 6­9:39 AM 2 5.3 Inverse Functions.notebook January 06, 2017 Jan 6­9:37 AM 3 5.3 Inverse Functions.notebook January 06, 2017 Learning Targets 1. Verify two functions are inverses 2. Find inverses of functions 3. Find the derivative of inverse functions Learning Target 4 5.3 Inverse Functions.notebook January 06, 2017 ∫ Definition of Inverse Function A function g is the inverse function of the function f if for each x in the domain of g and for each x in the domain of f. 1. Verify and are inverses. look at graph horizontal line test look at tables check to see if the slopes of the functions are reciprocal at (2, 11) and (11, 2). I.Definition of Inverse Function 5 5.3 Inverse Functions.notebook January 06, 2017 ∫∫ Finding Inverses The existence of an inverse function: 1. A function has an inverse function if and only if it is one‐to‐one. 2. If f is strictly monotonic on its entire domain, then it is one‐to‐one and therefore has an inverse funciton. *strictly monotonic means the function is either increasing on its entire domain or decreasing on its entire domain. 2. Do the functions below have an inverse? If so, find it (if possible). verify by derivative note domain II.Finding Inverses 6 5.3 Inverse Functions.notebook January 06, 2017 Jan 6­9:50 AM 7 5.3 Inverse Functions.notebook January 06, 2017 ∫∫ Finding Inverses 3. Show that the sine function is not one‐to‐one, then find the largest interval, centered at the origin, on which f is strictly monotonic. II.Finding Inverses 8 5.3 Inverse Functions.notebook January 06, 2017 ∫∫∫Derivatives of Inverse Functions *f(2) = 3 4. For the function a. What is the value of f ‐1(x) when x = 3? b. What is the value of (f ‐1)'(x) when x = 3? Using Theorem 5.9, now find f '(x), then find f ' (2) Note... III.Derivatives of Inverse Functions 9 5.3 Inverse Functions.notebook January 06, 2017 ∫∫∫Derivatives of Inverse Functions 5. Verify that has an inverse. Find III.Derivatives of Inverse Functions 10 5.3 Inverse Functions.notebook January 06, 2017 Your Homework AB Calculus: 5.3 p.349 #5,8,10,19,22,23,34,35,38,40,41,45,48, 54,57,63,67‐71,79,84,85,88,89,92,97, 100‐104 BC Calculus: 11.6 p.820 #3,8,15,17,40,44,47,59,63,69‐73 You know 11.1‐11.5 from pre‐caluculus, but good luck with 11.6! Homework 11.