algebra_of_functions

# 10 29 29 29 2 2 2 1 3 2 4 x x f g x 29 29 29 29 2 2 2

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10 ( 29 ( 29 ( 29 2 2 2 1 3 2 4 x x f g x = - = - - ( 29 29 ( 29 ( 29 2 2 2 2 1 2 6 9 1 2 12 18 1 3 g x x x f x x x - = - = - + - = - + - Example 3 Evaluate and : ( 29 ( 29 f g x o ( 29 ( 29 g f x o ( 29 3 f x x = - ( 29 2 2 1 g x x = - ( 29 ( 29 2 2 4 f g x x = - o ( 29 ( 29 2 2 12 17 g f x x x = - + o You can see that function composition is not commutative!

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©1999 by Design Science, Inc. 11 (Since a radicand can’t be negative in the set of real numbers, x must be greater than or equal to zero.) Example 4 Find the domain of and : ( 29 ( 29 f g x o ( 29 ( 29 g f x o ( 29 1 f x x = - ( 29 g x x = ( 29 ( 29 { } 1 : 0 f g f g x x D x x = - = o o ( 29 ( 29 { } 1 : 1 g f g f x x D x x = - = o o (Since a radicand can’t be negative in the set of real numbers, x – 1 must be greater than or equal to zero.)
©1999 by Design Science, Inc. 12 Example 5 The number of bicycle helmets produced in a factory each day is a function of the number of hours ( t ) the assembly line is in operation that day and is given by n = P ( t ) = 75 t – 2 t 2 . The cost C of producing the helmets is a function of the number of helmets produced and is given by C ( n ) = 7 n +1000 . Determine a function that gives the cost of producing the helmets in terms of the number of hours the assembly line is functioning on a given day. Find the cost of the bicycle helmets produced on a day when the assembly line was functioning 12 hours. (solution on next slide)

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