Square Root Function

A function f from

$\mathbb{R}$ to

$\mathbb{R}$ in which the variable appears in the radicand of a radical of order 2.

The square root function is sometimes called a radical function of order 2 or a root function of order 2.
The basic form of the rule of a square root function is

$f(x) = \sqrt{x}$ .
The standard form of the rule of the square root function is

$f(x) = a\sqrt{b(x − h)} + k$ where a is not equal to zero and b > 0.

The square root function is a particular case of an nth root function.

Example

Consider the function f(x)= $2\sqrt{0.5\left ( x – 3 \right )} + \left ( -1 \right )$

Educational notes

The square root function, also called a radical function of order 2, is derived from the reciprocal relation of the second-degree polynomial function. Although this reciprocal relation is not a function, two square root functions may be derived from it : one positive and one negative.

For example, if f is a polynomial function defined by the relation y = 0.5x $^{2}$ + 20, then the reciprocal relationship is obtained by interchanging the two variables: x = 0.5y $^{2}$ + 20. By isolating y, we obtain : y = ± $\sqrt{2x – 40}$ .