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Composition of Functions

We can flip this all around too! We've been finding( f o g )( x )
which isf( g( x ) )... 

But, if that wasn't exciting enough (try to calm down), we can find

( g o f )( x )

(Oh, boy!  Oh, boy!)

Check it out:

Givenf( x ) = 6x + 1andg( x ) = 2x^2 + 5
 

Let's find

 

( g o f )( x )

 

... which is

 

g( f( x ) )


       1 ) g( blob ) = 2( blob )^2 + 5

       2 ) g( f( x ) ) = 2( f( x ) )^2 + 5

       3 ) g( 6x + 1 ) = 2( 6x + 1 )^2 + 5 ... = 2( 36x^2 + 12x + 1 ) + 5 ... = 72x^2 + 24x + 7

We found( f o g )( x )before:

( f o g )( x ) = 12x^2 + 31

and( g o f )( x ) = 72x^2 + 24x + 7
 

Notice that

 

( f o g )( x ) does not equal ( g o f )( x )


!

Usually, you WILL get two different things.

That's why it's so important to read the notation the right way... or you may end up doing it backwards!

Preview:  If two functions are inverses of each other, you WILL get
                  the same thing for both!