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# Function Operations

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### Function Operations

1. 1. 6.3 Function Operations
2. 2. Review: What is a function? A relationship where every domain (x value has exactly one unique range (y value).  Sometimes we talk about a FUNCTION MACHINE, where a rule is applied to each input of x 
3. 3. Function Operations Addition Multiplica : f g ( x) tion : f g x Subtractio n : f Division f x : f g g x x f x g x f x g x g x f x g x where g x 0
4. 4. Adding and Subtracting Functions Let f x 3x Find f f 5x 2x 12 . f g ( x) g and f - g g ( x) (3 x 8 and g x 8) 4 f x (2 x g x 12 ) (3 x x 8) f x (2 x 20 When we look at functions we also want to look at their domains (valid x values). In this case, the domain is all real numbers. g x 12 )
5. 5. Multiplying Functions 2 Let f x x - 1 and g x x 1. Find f g f x x 3 g (x) x 2 (x x 2 1 1)( x 1) In this case, the domain is all real numbers because there are no values that will make the function invalid.
6. 6. Dividing Functions 2 Let f x x - 1 and g x x 1. f Find g f x x g x (x 2 x 1)( x (x 1) 1 1 1) x 1 In this case, the domain is all real numbers EXCEPT -1, because x=-1 would give a zero in the denominator.
7. 7. Let’s Try Some Let f x Find f x g ( x) 2 5 x - 1 and g x What is the domain? 5x 1. Find f x g ( x)
8. 8. Composite Function – When you combine two or more functions  The composition of function g with function is written as g f x g f x 1 1. Evaluate the inner function f(x) first. 2. Then use your answer as the input of the outer function g(x). 2
9. 9. Example – Composition of Functions Let f x x 2 and g x Method 1: g f g x g f x . Find 5 Method 2: x g f g f x g(x g f 2 2) 5 ( 7) (x 5 2 49 2) 2 2 2 g f x g f x 5 g( 5 g ( 7) ( 7) 2 49 2)