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Precalculus Section 2.6A Notes Name__________________________
Definitions: 1. ( f g )( ) x f ( ) x g x ( ) (add the two functions together)
2. ( f g )( ) x f ( ) x g x ( ) (subtract the two functions) 3. ( fg )( ) x f ( ) x g x ( ) (multiply the two functions)
f g x f^ x g x
(divide the functions as long as g x ( ) 0 )
Ex.: Find f g f , g fg f , , / g.
1. f ( ) x 2 x^3^ 3, g x ( ) 4 x^31 3 3 3
f g x x x x
3 3 6 3
fg x x x x x
3 3 3
f g x x x x
3 3
( / )( ) (2^ 3)
f g x x x (reduce if possible)
2. ( ) 2 1 , ( )^4
f x x^ g x x x x
f g x x^ x x x x x
2 2
fg x x^ x x x x x x
f g x x^ x x x x x
f g x x^ x x x x x x x x x
Domain of these functions: Must remove anything that makes either function undefined.
Definition: A composition of functions is written either f g x ( ( ))or ( f g )( ) x. In either case, g takes an input and makes an image based on its definition. f takes what g gives it and makes its own image based on its definition.
Ex: Evaluate the composite function in the following ways: f ( g (2)), g f ( (3)),( f f )(1),( g g )(0)
1. f ( ) x 3 x 2, g x ( ) 2 x^21
To evaluate, you work inside out.
***** f ( g (2)) Plug in 2 into the g function: g (2) 8 1 7
Then, plug the 7 into f : f (7) 23
So f ( g (2)) f (7) 23
***** g f ( (3)) Plug in 3 into the f function: f (3) 11
Then, plug 11 into g : g^ (11)^ 2(11)^21
So g f ( (3)) g (11) 241
***** ( f f )(1) This is another way of writing f ( f (1))
f ( f (1)) f (5) 17
***** ( g g )(0) g g ( (0)) g ( 1) 1
***** f g (3) Let’s do this one on the calculator. Enter f as Y1 and g as Y2 as
shown in the screen shot below:
Back on the home screen (2nd^ QUIT), enter the screen below (since Y1 is the outside
function and Y2 is the inside function):
