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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
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View Full Documentnever be negative and thus the range will be all positive numbers. Based oﬀ
this formula, the area
depends
on the radius, so
r
is the independent variable
and
A
is the dependent variable.
Another way to ﬁnd the domain is to think of the points you can’t plug in.
Try to ﬁnd the points where, if you were to plug them in, the function “breaks”.
This is usually most obvious when the function is represented as a formula.
Example 1.5.
a)
f
(
x
) =
√
x
+ 5, b)
g
(
x
) =
1
x
2

x
I say it’s more obvious because you can look at the function and you can
usually tell the basic idea of where it will break. You can use this to ﬁnd nice
formulas where it breaks. So what about those two?
a)
We know that we can only take the square root of a positive number. So
whatever is under the square root symbol has to be bigger than or equal to 0.
We write this as
x
+ 5
≥
0. We can solve this inequality to get
x
≥ 
5. We
write this as either
x
≥ 
5 or using the interval notation [

5
,
∞
)
b)
Well, with fractions, we know that we can’t divide by zero but anything
else is just ﬁne. So we’re only worried about where the denominator is 0. We
write this as
x
2

x
= 0, or
x
(
x

1) = 0. This is easy to solve and ﬁnd that
x
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 Spring '08
 varies
 Geometry

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