The topic with functions that we need to deal with is combining functions.
For the most part this means performing basic arithmetic (addition,
subtraction, multiplication, and division) with functions. There is one new way
of combing functions that we’ll need to look at as well.
Let’s start with basic arithmetic of functions. Given two functions
f (x) and g (x) we
have the following notation and operations.
(f + g)(x) = f (x) + g (x) (f
– g)(x) = f (x) – g (x)
(f g)(x) = f (x) g (x) 
Sometimes we will drop the (x) part
and just write the following,
f + g = f (x) + g (x) f – g = f (x) –
g (x)
f g = f (x) g (x) 
Note as well that we put x’s in the parenthesis, but we will
often put in numbers as well.
Now we need to discuss the new method of combining functions. The
new method of combining functions is called function composition. Here
is the definition.
Given two functions f (x) and g
(x) we
have the following two definitions.
( f ∘ g )(x) = f [ g (x) ]
2.
The composition
of g ( x) and f
( x) (again,
note the order) is,
(g ∘ f
)(x) = g [ f (x) ]
We need to note a couple of things here about function composition.
First this is NOT multiplication. Regardless of what the notation may suggest to you
this is simply not multiplication.
Second, the order we’ve listed the two functions is very important
since more often than not we will get different answers depending on the order
we’ve listed them.
Finally, function composition is really nothing more than function
evaluation. All we’re really doing is plugging the second function listed into the first function
listed. In the definitions we used [ ] for
the function evaluation instead of the standard ( ) to
avoid confusion with too many sets of parenthesis, but the evaluation will work
the same.
So, as we’ve seen from this last example it is possible to get the
same answer from both compositions on occasion. In fact when the answer from
both composition is x, as it is in this case, we know that these two
functions are very special functions. In fact, they are so special that we’re
going to devote the whole next section to these kinds of functions. So, let’s
move onto the next section.
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