### 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:

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) = 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 .