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 :

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*) = 2*x* + 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

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

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