1
Functions
1.1
What is a function?
All a function is, is something that takes a number and turns it into another
number.
Example 1.1.
Remember from geometry class the formula for a circle,
A
=
πr
2
.
This is a nice example of a function. It takes a number,
r
, and spits out another
number,
A
.
Example 1.2.
For a function, we don’t need a simple formula. The cost,
C
, of
mailing a letter that weights
w
follows a very speciﬁc formula, but it’s not as
nice as the formula for the area of a circle.
Example 1.3.
Functions don’t even need to have formulas. Think of the pop-
ulation of the world for a given year. This takes a number, the year, and spits
out another number, the population. There’s no real formula, however. And we
can’t plug in future years and expect to get back an answer that’s even close
Now for a better deﬁnition
Deﬁnition 1.
A
function
f
is a rule that assigns to each element
x
in a set
A
exactly
one element, called
f
(
x
), in a set
B
.
What does this mean? Look at this diagram:
The set
A
is called the
domain
of
f
and represents every value you can
plug into
f
and get out another number. Now, a function may not hit every
number in the set
B
. But the set of points that
f
(
x
) takes on is called the
range
of
f
. A symbol that represents an element of the domain is called the
independent variable. A symbol that represents an element in the domain is
called a dependent variable.
Example 1.4.
Let’s go back to the area of a circle,
A
=
πr
2
. Sure, we can
plug any number into
r
, but remember that it represents a length, so we’re not
going to be plugging in any negative numbers. Anything else goes, however, so
the domain is all positive real numbers. Also,
A
is an area, so it will certainly
1