Most of the functions we have studied in
Algebra I are defined
for all real numbers. This domain is denoted . For example, the domain of
f (x) = 2x + 5 is , because
f (x) is defined for all real numbers
x; that is, we can find
f (x) for all
real numbers
x. The domain of
f (x) = 
x2 - 6 is also , because
f (x) is defined for all real numbers
x.
Some functions, however, are not defined for all the real numbers, and thus are
evaluated over a
restricted domain. For example, the domain of
f (x) = 
is , because we cannot take the square root of a
negative number. The domain of
f (x) = 
is . The
domain of
f (x) = 
is , because we cannot divide by zero.
In general, there are two types of restrictions on domain: restrictions of an
infinite set of numbers, and restrictions of a few points. Square root signs
restrict an infinite set of numbers, because an infinite set of numbers make the
value under the
sign negative. To find the domain of a function with
a square root sign, set the expression under the sign greater than or equal to
zero, and solve for
x. For example, find the domain of
f (x) = 
- 11:
| 2x + 4 |
≥ |
0 |
|
| 2x |
≥ |
-4 |
|
| x |
≥ |
-2 |
|
The domain of
f (x) = 
- 11 is .
Rational expressions, on the other
hand, restrict only a few points, namely those which make the denominator equal
to zero. To find the domain of a function with a rational expression, set the
denominator of the expression not equal to zero and solve for
x using the zero
product property. For example, find the domain of
f (x) = 
:
| (x - 9)(2x + 8)(x + 2) |
≠ |
0 |
|
| x |
≠ |
9, - 4, - 2 |
|
The domain of
f (x) = is .
