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Daily Check Simplify: 2 5 2 (2 x y ) 1) 3x 2 y 5 z 4 2) 6 x3 yz 6 Math II UNIT QUESTION: What methods can be used to find the inverse of a function? Standard: MM2A2, MM2A5 Today’s Question: How do you find the composite of two functions and the resulting domain? Standard: MM2A5.d 4.2 Composition of Functions Objective To form and evaluate composite functions. To determine the domain for composite functions. Composition of functions Composition of functions is the successive application of the functions in a specific order. Given two functions f and g, the composite function f g is defined by f g x f g x and is read “f of g of x.” The domain of f g is the set of elements x in the domain of g such that g(x) is in the domain of f. Another way to say that is to say that “the range of function g must be in the domain of function f.” A composite function f g x g g(x) domain of g f range of f f(g(x)) range of g domain of f A different way to look at it… f g x x f g Function Machine gFunction x Machine Example 1 Evaluate f g x and g f x : f x x 3 g x 2x 2 gf fgxx 22 x 231 31 1 2 22 xx2246 x 9 1 2 x 2 12 x 18 1 f g x 2x 2 4 g f x 2x 2 12x 17 You can see that function composition is not commutative! Example 2 Evaluate f g x and g f x : f x 2x 3 2x fg gf x x 2x 1 3 3 1 3 2 x g x x 1 2 3 2x 3 x 2 f g x 3 x 1 g f x 3 2x 1 Again, not the same function. What is the domain??? Example 3 Find the domain of f g x and g f x : f x x 1 g x f x g x x 1 Df g x : x 0 (Since a radicand can’t be negative in the set of real numbers, f be x greater x than 1 or Dgequal x 1 xgmust f x to: zero.) (Since a radicand can’t be negative in the set of real numbers, x – 1 must be greater than or equal to zero.) Your turn Evaluate f g x and g f x : f x 3x 2 g x x 5 Example 4 Find the indicated values for the following functions if: f x 2x 3 2 g x x 1 f (g (1)) f (g (4)) g (f (2)) g (g (2)) 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) = 75t – 2t2. The cost C of producing the helmets is a function of the number of helmets produced and is given by C(n) = 7n +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) n P t 75t 2t 2 C n 7n 1000 Solution to Example 5: 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. Cost C n C P t 75t 2t 1000 7 75 C 75t 2t 2 2 14t 2 525t 1000 Find the cost of the bicycle helmets produced on a day when the assembly line was functioning 12 hours. C 14t 2 525t 1000 $5284 Summary… Function arithmetic – add the functions (subtract, etc) Addition Subtraction Multiplication Division Function composition Perform function in innermost parentheses first Domain of “main” function must include range of “inner” function Class work Workbook Page 123-124 #13-24 Homework Page 114 #19-24 Page 115 #9-16