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Daily Check Simplify: 1) 2)

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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

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4.2 Composition of Functions

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Objective To form and evaluate composite functions. To determine the domain for composite functions.

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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 is defined by and is read “f of g of x.” The domain of 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.”

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A composite function x g(x)g(x) f(g(x)) domain of g range of f range of g domain of f g f

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A different way to look at it… Function Machine x Function Machine g f

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Example 1 Evaluate and : You can see that function composition is not commutative!

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Example 2 Evaluate and : Again, not the same function. What is the domain???

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(Since a radicand can’t be negative in the set of real numbers, x must be greater than or equal to zero.) Example 3 Find the domain of and : (Since a radicand can’t be negative in the set of real numbers, x – 1 must be greater than or equal to zero.)

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Your turn Evaluate and :

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Example 4 Find the indicated values for the following functions if:

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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 – 2t 2. 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)

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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 to Example 5:

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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

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Class work Workbook Page 123-124 #13-24

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Homework Page 114 #19-24 Page 115 #9-16

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5.1 Composite Functions Goals 1.Form f(g(x)) = (f g) (x) 2.Show that 2 Composites are Equal.

5.1 Composite Functions Goals 1.Form f(g(x)) = (f g) (x) 2.Show that 2 Composites are Equal.

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