Algebra II: Functions
Domain
Domain
The domain of a relation (or of a function) is the set of all inputs
of that relation. For example, the domain of the relation
(0, 1),(1, 2),(1, 3),(4, 6)
is
x=0, 1, 4
.
The domain of the following mapping diagram is
-2, 3, 4, 10
:
Mapping Diagram
The domain of the following graph is :
Graph
Restrictions on Domain
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) = 
x
2 - 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 .
