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:

The domain of the following graph is :

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) = :