1
MA 15200
Lesson 25
Section 2.6
I
The Domain of a Function
Remember that the domain is the set of
x
’s in a function, or the set of ‘first things’.
For
many functions, such as
( )
2
3
f x
x
=

,
x
could be replaced with any real number to find
( )
f x
.
Therefore the domain could be written in setbuilder notation as
{  is a real number}
x x
or (
, )
∞ ∞
using interval notation.
However, not all functions
have domains that are all real numbers.
In real life examples, the domain is often
limited
to numbers that only make sense.
For example, if
50
d
t
=
(distance in miles equals 50
times time in hours), the time can only be 0 or positive numbers.
No one would travel
negative hours.
So the domain would be { 
0} or [0, )
t t
≥
∞
.
Examine the following two functions.
2
( )
4
( )
3
f x
x
g x
x
=

=

If
x
is replaced with a 5 in function value for
f
, the result would be
1

.
While we’ve
discussed imaginary numbers, when speaking of domains of functions, we only think of
real numbers.
So we know 5 is not in the domain of
f
.
How do we find the domain?
We
know that we can only take the square root of a positive number or zero.
Therefore 4 –
x
must be greater than or equal to zero.
4
0
4
4
x
x
x

≥
 ≥ 
≤
The domain of
f
is { 
4} or (
,4]
x x
≤
∞
.
In function
g
, if
x
is replaced with 3, the result would equal
2
0
, which is an undefined
number.
3 is not in the domain of function
g
.
We can write the domain as
{  is a real number,
3} or (
,3)
(3, ).
x x
x
≠
∞
∞
∪
The above examples illustrate the ‘problems’ you will encounter possibly when
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 Fall '09
 Continuous function, Complex number, codomain

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