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 
and we have the following notation and operations.
Sometimes we will drop the part and just write the following,
Note as well that we put x’s
in the parenthesis, but we will often put in numbers as well. Let’s take a quick look at an example.
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Example 1 Given
and evaluate each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
By evaluate we mean one of two things depending on what is
in the parenthesis. If there is a
number in the parenthesis then we want a number. If there is an x (or no parenthesis, since that implies and x) then we will perform the operation and simplify as much as
possible.
(a)
In this case we’ve got a number so we need to do some
function evaluation.
[Return to Problems]
(b)
Here we don’t have an x
or a number so this implies the same thing as if there were an x in parenthesis. Therefore, we’ll subtract the two functions
and simplify. Note as well that this
is written in the opposite order from the definitions above, but it works the
same way.
[Return to Problems]
(c)
As with the last part this has an x in the parenthesis so we’ll multiply and then simplify.
[Return to Problems]
(d)
In this final part we’ve got a number so we’ll once again
be doing function evaluation.
[Return to Problems]
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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 and we have the following two definitions.
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.
Let’s take a look at a couple of examples.
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Example 2 Given
and evaluate each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a) These are the same functions that we used in the first set of
examples and we’ve already done this part there so we won’t redo all the work
here. It is here only here to prove
the point that function composition is NOT function multiplication.
Here is the multiplication of these two functions.
[Return to Problems]
(b) Now, for function composition all you need to remember is
that we are going to plug the second function listed into the first function
listed. If you can remember that you
should always be able to write down the basic formula for the composition.
Here is this function composition.
Now, notice that since we’ve got a formula for we went ahead and plugged that in
first. Also, we did this kind of
function evaluation in the first section we
looked at for functions. At the time
it probably didn’t seem all that useful to be looking at that kind of
function evaluation, yet here it is.
Let’s finish this problem out.
Notice that this is very different from the
multiplication! Remember that function
composition is NOT function multiplication.
[Return to Problems]
(c) We’ll not put in the detail in this part as it works
essentially the same as the previous part.
Notice that this is NOT the same answer as that from the
second part. In most cases the order
in which we do the function composition will give different answers.
[Return to Problems]
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The ideas from the previous example are important enough to
make again. First, function composition
is NOT function multiplication. Second,
the order in which we do function composition is important. In most case we will get different answers
with a different order. Note however,
that there are times when we will get the same answer regardless of the order.
Let’s work a couple more examples.
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Example 3 Given
and evaluate each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
(a)
Not much to do here other than run through the formula.
[Return to Problems]
(b)
Again, not much to do here.
[Return to Problems]
(c)
Now in this case do not get excited about the fact that
the two functions here are the same.
Composition works the same way.
[Return to Problems]
(d)
In this case, unlike all the previous examples, we’ve got
a number in the parenthesis instead of an x,
but it works in exactly the same manner.
[Return to Problems]
(e)
Again, we’ve got a number here. This time there are actually two ways to do
this evaluation. The first is to
simply use the results from the first part since that is a formula for the
general function composition.
If we do the problem that way we get,
We could also do the evaluation directly as we did in the
previous part. The answers should be
the same regardless of how we get them.
So, to get another example down of this kind of evaluation let’s also
do the evaluation directly.
So, sure enough we got the same answer, although it did
take more work to get it.
[Return to Problems]
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Example 4 Given
and evaluate each of the following.
(a) [Solution]
(b) [Solution]
Solution
(a) Hopefully, by this point these aren’t too bad.
Looks like things simplified down considerable here.
[Return to Problems]
(b) All we need to do here is use the formula so let’s do that.
So, in this case we get the same answer regardless of the
order we did the composition in.
[Return to Problems]
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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.