Irrational functions

An irrational function is a function whose analytic expression has the independent variable $x$ under the root symbol.

In this paragraph we will consider only irrational functions of the type $$\displaystyle f(x)=\sqrt[n]{g(x)}$$ with $g(x)$ a rational function.

• If the index $n$ of the root is odd, it is possible to calculate the image of any real number, if the expression $g (x)$ is a real number, that is $Dom(f)=Dom(g)$.
• If the index $n$ of the root is a even, to be able to calculate images we need $g (x)$ to be positive or zero, since the even roots of a negative number are not real numbers. Therefore the domain of $f$ are the solutions of the inequation $g(x) \geq 0$. In other words, $Dom (f) = \{x \in \mathbb{R} \mid g(x) \geq 0\}$.

Let's study now the simplest case of irrational function: the square root function $\displaystyle f(x)=\sqrt{x}$.

This is a function in which the index of the root is $2$. Therefore, its domain is the set of solutions of the inequation $x \geq 0$. Thus we have $Dom (f) = [0, +\infty)$ The image of the square root function is, as is the case of the domain, the set of the positive numbers, $Im (f) = [0, +\infty)$

Let's see its graphic representation: