Rational functions

A rational function is a function whose analytic expression is given by a quotient between polynomials: $${\displaystyle f(x)=\frac{P(x)}{Q(x)}}$$ In this type of function it is possible to calculate the image of any real number except those that are solutions of the denominator, since we will have to divide by $0$ and thus we do not obtain a real number.

Therefore we can define the domain of this type of functions by $$Dom(f)=\mathbb{R}-\{x\in \mathbb{R}\mid Q(x)=0\}$$ An important case of rational function is the function of inverse proportionality: ${\displaystyle f(x)=\frac{k}{x}}$ where $k$ is a constant. Note that it is a rational function with $P(x)=k\ne 0$ and $Q(x)=x$.

Its domain is the set of the real numbers that are not solutions to the denominator, that is, $Dom(f)=\mathbb{R}-\{0\}$

Its image is the set of real numbers except zero, since it is not an image of any element of the domain; that is, $Im(f)=\mathbb{R}-\{0\}$

Let's see the graph of the function ${\displaystyle f(x)=\frac{1}{x}}$: