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What is a function?
The idea for this problem sequence was suggested by Michelle Manes
In algebra we often talk about relationships between variables.
We also talk about functions. Here are some questions to
think about: Are relationships and functions the same things?
Do relationships and functions exist only in algebra? The
following problems help to answer these questions. The sidenote
reminds you how to distinguish between a function and a
relationship.
Some
people
think
of
a
function
as
a
machine
that
takes
an
input
and
produces
an
output.
An
input
can
be
of
any
sort:
a
number,
a
geometric
shape,
a
person,
a
country,
or
anything
else.
But
for
a
particular
input
there
is
always
a
particular
output.
For
example,
if
the
input
is
a
number,
and
the
output
is
twice
a
number,
this
is
a
function.
Another
example
of
a
function
would
be:
the
input
is
a
point
on
a
coordinate
plane
and
the
output
is
a
set
of
x-
and
y-coordinates
for
this
point.
If
there
are
two
or
more
different
outputs
possible
for
one
input,
we
are
talking
about
a
relationship,
but
not
a
function.
An
example
of
a
relationship
that
is
not
a
function
is
an
input
is
a
number
and
an
output
that
is
a
factor
of
this
number.
The
set
of
allowable
inputs
to
a
function
is
called
the
domain
of
that
function.
The
set
of
all
outputs
of
the
function
is
called
the
range
of
the
function.
For each of the following examples decide whether it is a function,
and explain your decision. For each example that is a function,
describe the domain and the range.
- Input: a number
Output: seven more than that number
- Input: a whole number
Output: whether the number is even or odd
- Input: a country
Output: the capital of that country today
- Input: a person
Output: the name of that person’s cousin
- Input: the radius of a circle
Output: the area of that circle
- Input: the length of one side of a triangle
Output: the perimeter of that triangle
- Input: a person
Output: the height of that person
- Input: a number
Output: always the number 186
- Create one more example of a function and one more
example of a relationship that is not a function.
Hints
Hint to problem 7. You can think of the two different situations.
One is, you measure the height of many people just once. Another
one is, you measure the heights of different people a number of
times over a number of years.
Hint to problem 8. Think about each input-output pair. Is there
a single output for a given input?
Answers
See solutions.
Solutions
- It is a function, because if x is a number (an input), there
is only one number which can be expressed as x + 7 (an
output). The domain of this function is all real numbers.
The range is also all real numbers.
- It is a function, because the number is either even or odd,
there is only one output for every input. The domain of
this function is all whole numbers. The range is the two
categories: even and odd.
- It is a function, because for every country, there is only
one capital. The domain for this function is all countries
in the world. The range for this function is all capitals of
the countries in the world.
- This is not a function. A person can have more than one
cousin with more than one name.
- This is a function. A circle with a given radius has only one
area. The domain for this function is any positive number.
The range is also any positive number.
- This is not a function. If only one side of the triangle is
given, other sides can vary, and there could be various
perimeters.
- You can consider it a function or not a function depending
on how you look at it. If you think about the height of
a person throughout that person’s life, then it is not a
function, because for each person there are many possible
heights. If you consider many different people at a single
point in time, then it is a function. At a certain point in
time every person has a unique height. If you consider it
as a function, the domain is all people. The range is all
possible heights these people have.
You
can
say
that
the
height
is
a
function
of
the
two
variables:
person
and
time.
If
the
person
is
the
same,
but
the
time
varies,
it
is
a
function
of
time.
If
the
time
is
the
same,
but
the
person
varies,
it
is
the
function
of
a
person.
- It is a function, because for any number, you have only
one output, that is the number 186. The domain of this
function is all real numbers. The range consists of the single
number 186.
- Answers will vary.
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