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Mathematical Expression Editor
We discuss compositions of functions.
Given two functions, we can compose them. Let’s give an example in a ‘‘real
context.’’
Let and let What does represent in this setting?
With we first relate how
far one can drive with gallons of gas, and this in turn is determined by
how much money one has. Hence represents how far one can drive with
dollars.
Composition of functions can be thought of as putting one function inside another.
We use the notation
The composition only makes sense if
Suppose we have
Find and state its domain.
The range of is , which is equal to the domain of . This
means the domain of is . Next, we substitute for each instance of found in and so
Now let’s try an example with a more restricted domain.
Suppose we have:
Find and state its domain.
The domain of is . From this we can see that the range
of is . This is contained in the domain of .
This means that the domain of is . Next, we substitute for each instance of found
in and so
We can summarize our results as a piecewise function, which looks somewhat
interesting:
Suppose we have:
Find and state its domain.
While the domain of is , its range is only . This is
exactly the domain of . This means that the domain of is . Now we may substitute
for each instance of found in and so
Compare and contrast the previous two examples. We used the same functions for
each example, but composed them in different ways. The resulting compositions are
not only different, they have different domains!
Many functions that you will encounter in math are actually a composition of
functions, and you will need to recognize which functions they are composed of. Let’s
practice this.
The function is a composition of which of the following functions? (Select all that
apply)
Now that you know is composed of and , choose which composition results in
.
The function is a composition of which of the following functions? (Select all that
apply)
Hint:
Now that you know is composed of , , and , choose which composition results in
.
The function is a composition of which of the following functions? (Select all that
apply)
Now that you know is composed of , , and , choose which composition results in
.