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Finding Domains of
Composite Functions
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At times, the domain of a composite function can be a bit
confusing. Let's examine
what happens to values as they "travel" through a composition of
functions.
Consider the following example:
and
What is the domain of
 ?
In this problem, function
 cannot
pick up the value x = 3,
and function
cannot pick up the value x = -2.
The domain of
will be the values from the domain of
which can "move through" to the end of the composition. This means
that
the answers created by these values from function
must be "picked up"
by function.
Let's follow this
process algebraically:
1. Function
cannot pick up the value 3. Consequently, the composition also
cannot pick up the value 3.
2. The answers coming out of function
come out in the form
 .
Since function
cannot pick up -2, we must lookout for any values of x that cause
since these values create an answer that cannot
progress through the composition
(cannot be picked up by function
).
3. When does
? Solve algebraically ....
4. The domain of
will be all real numbers with the exclusion of 3 and
2.
(Notice that one of the excluded
values is 2, not -2. The value x = -2 makes it through
the composition very nicely because its
answer from function
is 2/5 which is
then picked up by function
.)
Is there an easier way
to find the domain of a composition?
If you are finding the algebraic
expression for the composition of two functions, you can examine your
answer to determine any additional restrictions on the domain of the
composition. Let's continue with
our problem....
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The algebraic expression for this
composition SHOWS that x = 2 would not be an acceptable
domain element since it creates a zero denominator problem.
Just remember that
you must also
specify any restrictions on the domain of the starting function.
In this problem, x = 3 is not allowed
since it is a restriction on
.
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Answer: The
domain of the composition is all real numbers with the exclusion of 3 and
2.
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