When we compose functions, we take a function of a function. For example, suppose the temperature
on a given day is described as a function of time
(measured in hours after midnight) as in
[link] . Suppose the cost
to heat or cool a building for 1 hour, can be described as a function of the temperature
Combining these two functions, we can describe the cost of heating or cooling a building as a function of time by evaluating
We have defined a new function, denoted
which is defined such that
for all
in the domain of
This new function is called a composite function. We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function
It does not make sense to consider
because temperature is not a function of cost.
Definition
Consider the function
with domain
and range
and the function
with domain
and range
If
is a subset of
then the
composite function
is the function with domain
such that
A composite function
can be viewed in two steps. First, the function
maps each input
in the domain of
to its output
in the range of
Second, since the range of
is a subset of the domain of
the output
is an element in the domain of
and therefore it is mapped to an output
in the range of
In
[link] , we see a visual image of a composite function.
For the composite function
we have
and
Compositions of functions defined by formulas
Consider the functions
and
Find
and state its domain and range.
Evaluate
Find
and state its domain and range.
Evaluate
We can find the formula for
in two different ways. We could write
Alternatively, we could write
Since
for all real numbers
the domain of
is the set of all real numbers. Since
the range is, at most, the interval
To show that the range is this entire interval, we let
and solve this equation for
to show that for all
in the interval
there exists a real number
such that
Solving this equation for
we see that
which implies that
If
is in the interval
the expression under the radical is nonnegative, and therefore there exists a real number
such that
We conclude that the range of
is the interval
We can find a formula for
in two ways. First, we could write
Alternatively, we could write
The domain of
is the set of all real numbers
such that
To find the range of
we need to find all values
for which there exists a real number
such that
Solving this equation for
we see that we need
to satisfy
which simplifies to
Finally, we obtain
Since
is a real number if and only if
the range of
is the set
