4.8 Rational function

We call rational function any function of the form

\begin{equation*} f(x) = \frac{P(x)}{Q(x)}, \end{equation*}

where $P$ and $Q$ are polynomials. Generally speaking, the domain of such a function is given by the set of all real numbers that does not contain roots of the polynomial $Q$, i.e.

\begin{equation*} D_f = \{ x \in \mathbb{R} \mid Q(x) \neq 0 \}. \end{equation*}

Rational functions include linear, quadratic and all polynomial functions. Simply, you just have to set $Q(x) = 1$, for $x\in\mathbb{R}$ and $P$ to be any polynomial.

It is no longer easy to say something about the image set, so we will not dicuss this question. However, let us at least show a few examples illustrating that there can be very diverse situations (see picture 4.9).

Figure 4.9: Examples of rational functions.