5_3_inverse_functions

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Unformatted text preview: 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Introduction (Additional practice for 5.2: p. 338 #13, 15, 17, 41) HW: 5.3: p. 347 #1, 5, 29, 33, 37, 41, 47, 51, 59, 77, 79, 2325 Consider and compare: f(x) = 4 x + 3 and g(x) = x 3 4 Start with x. Then: Note the inverse operations Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Intro Prof. Battaly, WCC, Calculus 1 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Introduction (Additional practice for 5.2: p. 338 #13, 15, 17, 41) HW: 5.3: p. 347 #1, 5, 29, 33, 37, 41, 47, 51, 59, 77, 79, 2325 Consider and compare: f(x) = 4 x + 3 and g(x) = x 3 4 Start with x. Then: Note the inverse operations Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Intro Prof. Battaly, WCC, Calculus 2 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Introduction Consider and compare: 2 f(x) = (2 x 3 ) and g(x) = x + 3 2 Start with x. Then: These appear to be inverse functions, but what about x=0? Are they inverse functions for all x? all y? Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Intro Prof. Battaly, WCC, Calculus 3 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Introduction Consider and compare: 2 f(x) = (2 x 3 ) and g(x) = x + 3 2 Start with x. Then: These appear to be inverse functions, but what about x=0? Are they inverse functions for all x? all y? Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Intro Prof. Battaly, WCC, Calculus 4 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Introduction 2 f(x) = (2 x 3 ) , x 3/2 and g(x) = x + 3 , x 0 2 But, f and g are inverses ONLY when x 0 for g(x) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Graph Preview Prof. Battaly, WCC, Calculus 5 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Definition A function g is the inverse of a function f if f(g(x)) = x for each x in the domain of g AND g(f(x)) = x for each x in the domain of f (f 1 notation means "f inverse", NOT exponent) and g(x) = f 1 (x) Notes 1 1 1. If g(x) = f (x), then f(x) = g (x) 2. Domain of f 1 is the range of f and the range of f 1 is the domain of f 3. If f 1 exists, then it is unique (only one) 4. goes in both directions Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 definition Prof. Battaly, WCC, Calculus 6 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 example Prof. Battaly, WCC, Calculus 7 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 example Prof. Battaly, WCC, Calculus 8 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 example Prof. Battaly, WCC, Calculus 9 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions Explore relationships between a function and its inverse, using Geogebra. Click on the globe. Uses a quadratic function. Explore relationships between a function and its inverse, using Geogebra. Click on the globe. Uses ln x. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 example Prof. Battaly, WCC, Calculus 10 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Properties (Insert images from APCD or demo the APCD) skip if have done geogebra for ln x Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 properties Prof. Battaly, WCC, Calculus 11 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Properties (Insert images from APCD or demo the APCD) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 properties Prof. Battaly, WCC, Calculus 12 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Properties (Insert images from APCD or demo the APCD) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 properties Prof. Battaly, WCC, Calculus 13 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Properties (Insert images from APCD or demo the APCD) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 properties Prof. Battaly, WCC, Calculus 14 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Properties (Insert images from APCD or demo the APCD) (a,b) on f slope of f (b,a) on f slope of f1 1 Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 points and slopes Prof. Battaly, WCC, Calculus 15 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Reflective Property of Inverse Function The graph of f contains the point (a,b) IFF the graph of f 1 contains the point (b,a) Derivative of an Inverse Function Let f be a function differentiable on I, and g(x) = f 1 (x) 1. Then g is differentiable at any x for which f ' (g(x)) 0 2. g ' (x) = 1 and f ' (g(x)) 0 f ' (g(x)) Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Reflective Prop & Derivative Prof. Battaly, WCC, Calculus 16 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Cannot have inverse over whole domain. Is strictly monotonic. Is 1to1. Can have inverse over whole domain. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 Need 1to1 Prof. Battaly, WCC, Calculus 17 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Exchange variables, and solve for y. Can check to be sure, using definition. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 example Prof. Battaly, WCC, Calculus 18 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: How do we restrict the domain for these to be inverse functions? Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 example Prof. Battaly, WCC, Calculus 19 5_3_inverse_functions.notebook February 04, 2009 5.3 Inverse Functions: Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework Part 1 Homework Part 2 example Prof. Battaly, WCC, Calculus 20 ...
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