Mastering Functions: Operations and Compositions
Dive into the world of functions by learning how to perform various operations and find composite functions. Explore real number iterations and discover the composition of functions through practical examples. Get a solid grasp on functions' essentials and boost your mathematical skills!
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Presentation Transcript
LESSON 1-2 COMPOSITION OF FUNCTIONS LEARNING OBJECTIVE(S): I CAN PERFORM OPERATIONS WITH FUNCTIONS (+, -, X, & ). I CAN FIND COMPOSITE FUNCTIONS. I CAN ITERATE FUNCTIONS USING REAL NUMBERS. ESSENTIAL QUESTION: How do I find the composition of functions?
EX.1 PERFORMING OPERATIONS WITH FUNCTIONS Operations with Functions: Sum: Difference: (f - g)(x) = f(x) - g(x) Product: (f g)(x) = f(x)g(x) Quotient: (f g)(x) = f(x) g(x), where g(x) 0 (f + g)(x) = f(x) + g(x)
EX.1 PERFORMING OPERATIONS WITH FUNCTIONS I/WE DO: Given ? ? = 3?2 4 ??? ? ? = 4? + 5,???? ??? ????????. a.) ? ? + ?(?) b.) ? ? ?(?) c.) ? ? ?(?) d.) ? ? ?(?)
EX.1 PERFORMING OPERATIONS WITH FUNCTIONS I/WE DO: Given ? ? = 5?2+ 1 ??? ? ? = 2? + 5,???? ??? ????????. a.) ? ? + ?(?) b.) ? ? ?(?) c.) ? ? ?(?) d.) ? ? ?(?)
EX.1 PERFORMING OPERATIONS WITH FUNCTIONS YOU DO: Given ? ? = 2? 1 ??? ? ? = ?2,???? ??? ????????. a.) ? ? + ?(?) b.) ? ? ?(?) c.) ? ? ?(?) d.) ? ? ?(?)
EX.2 FINDING THE COMPOSITION OF FUNCTIONS Composition of Functions: Given functions f and g, the composite function (f g) can be described by the following equation: ( )( ) f g x = ( ( )) f g x f g The domain of includes all of the elements of x in the domain of g for which g(x) is in the domain of f.
EX.2 FINDING THE COMPOSITION OF FUNCTIONS I/WE DO: ????? ? ? =1 ?,? ? = ? + 7,& ? = 2 ?;Find the following Compositions: a.) f(g(2)) b.) h(f(x)) c.) g h(9) d.) g(f(x)) e.) f g and state the domain.
EX.2 FINDING THE COMPOSITION OF FUNCTIONS YOU DO: ????? ? ? = 2?2 3? + 8 ??? ? ? = 5? 6;Find the following Compositions: a.) f(g(2)) b.) g(f(x)) c.) f g(9) d.) f g and state the domain
EX.3 FINDING ITERATIONS OF A FUNCTION I/WE DO: Find the first three iterates,?1, ?2,??? ?3,?? ? ? ???????? ? ? = ?2+ 1 ??? ?? ??????? value of ?0= 2.
EX.3 FINDING ITERATIONS OF A FUNCTION YOU DO: Find the first three iterates,?1, ?2,??? ?3,?? ? ? ???????? ? ? = 2? 3 ??? ?? ??????? value of ?0= 1.
