Implied domain

When a function  is defined, it is often the case that the domain  of  is not explicitly stated. In such a case we will consider the domain  to be the set of real numbers  so that the value is also a real number. This is called the implied domain of . For example, is the implied domain of square root function  .

Equations of a semicircle. By the Pythagorean theorem the point  on the semicircle with radius  must satisfy

 (1)

By solving (1) in terms of  we obtain . Considering a semicircle for , we obtain an equation of the semicircle as follows.

Square root function. A function  is called a square root function if

Department of Mathematics