Precalculus Review

Implied Domain

Implied domain. When a function

is defined, it is often the case that the domain

is not explicitly stated. In this case the domain

will be the collection of real numbers

where the value

is also a real number. This is called the implied domain of

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

$\displaystyle f(x) = \sqrt{x}.$

Here the implied domain should be $D = [0,\infty)$ .

$\includegraphics{lec03b.ps}$

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

$\displaystyle x^2 + y^2 = r^2$

By solving the above equation in terms of

we obtain $y = \pm \sqrt{r^2 - x^2}$ . Considering a semicircle for $y \ge 0$ , we obtain an equation of the semicircle as follows.

$\displaystyle y = \sqrt{r^2 - x^2} .$

Here the implied domain is

$\includegraphics{lec03a.ps}$