Function Composition

WELCOME EVERYBODY! ^^ TODAY I'LL POST ABOUT FUNCTION COMPOSITION, HIGHSCHOOL GRADE. THIS MATERIAL IS SO CHALLENGING. BUT YOU CAN GET USED TO IT! ^^
 

The Basic Characteristic Of Function Compositions

       I.          Composition Function is :

( f o g ) (x) =  f( g(x) )  = f composition g for x

Example ^^ :

1.  f(x) = 5x – 4 and g(x) = 2x + 8

( f o g )? = f( g(x) ) 

f( 2x + 8 )  ----->  5( 2x + 8 ) – 4 ---> 10x + 40 – 4 ----> 10x + 36

as you can see above ...

Basically change the X value of F(x) into g(x) so become f( g(x) ) = f( 2x + 8 )   and because the F(x) = 5x – 4 logically F( 2x + 8) = 5( 2x + 8) – 4     ^^   ----->   10x + 8

How about ( g o f )?

Value still the same : f(x) = 5x – 4 and g(x) = 2x + 8

( g o f ) =  g( f(x) )

g(5x -4) ------>  2(5x – 4) + 8   ----->  10x – 8 + 8 = 10x    ^^

The answer is different so this is mean ( f o g )  IS NOT THE SAME AS ( g o f )

2.      F(x) = x² + 2x – 3 and g(x) = 5x -1

( f o g )? = f( g(x) ) 

f(5x – 1) ------>  (5x – 1)² + 2(5x – 1) – 3

               ------->  25x – 10x + 1 + 10x – 2 - 3

               ------->  (25x – 4) the answer!

How about ( g o f )(-1) ?

Value still the same :  F(x) = x² + 2x – 3 and g(x) = 5x - 1

( g o f ) =  g( f(x) ) ---> ( g o f )(-1) = g( f(-1) )

5(x² + 2x – 3) – 1 ----->  What was asked is ( g o f )(-1) . so change the x from ( f(x) )into value -1 FIRST like this  -----> 5((-1)² + 2(-1) – 3) – 1 

---> 5(1 -2 – 3) – 1

---> 5(-4) – 1 = -21 the answer!

Here some other Characteristic ^^   :

1.  Function Compositions is NOT commutative 

           ( f o g )  IS NOT THE SAME AS ( g o f )

     2.  Function Compositions IS Associative

            ( ( f o g ) o h)(x)  =  ( f o (g o h ) )(x)

     3.  There is Identity Function I(x) = x

           so ---> ( f o I )(x) = ( I o f )(x) = f(x)