When the output of one function is used as the input of another, we call the entire operation a composition of functions. For any input
and functions
and
this action defines a
composite function , which we write as
such that
The domain of the composite function
is all
such that
is in the domain of
and
is in the domain of
It is important to realize that the product of functions
is not the same as the function composition
because, in general,
Determining whether composition of functions is commutative
Using the functions provided, find
and
Determine whether the composition of the functions is
commutative .
Let’s begin by substituting
into
Now we can substitute
into
We find that
so the operation of function composition is not commutative.
The function
gives the number of calories burned completing
sit-ups, and
gives the number of sit-ups a person can complete in
minutes. Interpret
The inside expression in the composition is
Because the input to the
s -function is time,
represents 3 minutes, and
is the number of sit-ups completed in 3 minutes.
Using
as the input to the function
gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).
Suppose
gives miles that can be driven in
hours and
gives the gallons of gas used in driving
miles. Which of these expressions is meaningful:
or
The function
is a function whose output is the number of miles driven corresponding to the number of hours driven.
The function
is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:
The expression
takes miles as the input and a number of gallons as the output. The function
requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression
is meaningless.
The expression
takes hours as input and a number of miles driven as the output. The function
requires a number of miles as the input. Using
(miles driven) as an input value for
where gallons of gas depends on miles driven, does make sense. The expression
makes sense, and will yield the number of gallons of gas used,
driving a certain number of miles,
in
hours.
Are there any situations where
And
Would both be meaningful or useful expressions?
Yes. For many pure mathematical functions, both compositions make sense, even though they usually produce different new functions. In real-world problems, functions whose inputs and outputs have the same units also may give compositions that are meaningful in either order.
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
