Implied domain

When a function $f$ is defined, it is often the case that the domain $ D$ of $f$ is not explicitly stated. In such a case we will consider the domain $ D$ to be the set of real numbers $x$ so that the value $f(x)$ is also a real number. This is called the implied domain of $f$. For example, $ D = [0,\infty)$ is the implied domain of square root function  $ f(x) = \sqrt{x}$.

Equations of a semicircle. By the Pythagorean theorem the point $ (x,y)$ on the semicircle with radius $ r > 0$ must satisfy

$\displaystyle x^2 + y^2 = r^2$ (1)

By solving (1) in terms of $x$ 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} .


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

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


Department of Mathematics
Last modified: 2005-09-29