Given f(x)= 2+3x-x^{2 }and g(x) =2x-1 evaluate ( fg ) ( x ) , (fog)(x) and (gof )(x)
Solution
These are the similar functions that we utilized in the first set of instances and we've already done this part there hence we won't redo all the work here. This is here only here to demonstrate the point that function composition is not function multiplication.
Following is the multiplication of these two functions.
( fg ) ( x ) = -2 x^{3} + 7 x^{2} + x - 2
(b) Now, for the purpose of function composition all you have to remember is that we are going to plug the second function tabulated into the first function listed. If you can recall that you must always be capable to write down the fundamental formula for the composition.
Here is this function composition.
( f o g ) ( x ) = f [g ( x )]
= f [2x -1]
= 2+ 3( 2 x -1) - ( 2 x -1)^{2}
= 2 + 6 x - 3 - (4 x^{2} - 4 x + 1)
= -1 + 6 x - 4 x^{2} + 4 x -1
= -4 x^{2} + 10 x - 2
Notice that it is very different from the multiplication! Remember that function composition is NOT function multiplication.
c) We'll not put in the detail in this part as it works really the same as the previous part.
( g o f ) ( x ) = g [f ( x )]
= g[2+3x-x^{2}]
= 2(2+3x-x^{2})-1
=4+6x-2x^{2}-1
=-2x^{2}+6x+3
Notice that it is NOT the similar answer as that from the second part. In most of the cases the order in which we do the function composition will give distinct answers.
Some facts are important enough to make again. First, function composition is different from function multiplication. Second, the order of function composition is important. In most of the case we will get distinct answers with a different order. Note though, that there are times when we will get the same answer regardless of the order.