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# Rational functions

A rational function is the ratio of two polynomials $$f(x) = \frac{P(x)}{Q(x)}$$ It can always be in reduced form, i.e. with distinct zeros for $P$ and $Q$.

The degree $\operatorname{\textrm{deg}}(P)$ of a polynomial $P$ is the highest power of $P$.

The degree of $x+1$ is $1$ and the degree of $x^3-1$ is $3$.

The main properties of a rational function in reduced form are as follows.

• Domain. Everything but the zeros of $Q$.
• Vertical asymptote at each zero of $Q$.
• Zeros are the zeros of $P$.
• Horizontal and oblique asymptotes if $\operatorname{\textrm{deg}}(P) \le \operatorname{\textrm{deg}}(Q) + 1$. In this case, we write $f$ in the form $$ax + b + \frac{R(x)}{Q(x)}$$ where $\operatorname{\textrm{deg}}(R) \le\operatorname{\textrm{deg}}(Q)-1$. The asymptote is $y = ax+b$.

$$f(x) = \frac{x^3-x}{x^2-4}= \frac{x^3-4x+3x}{x^2-4} = x + \frac{ 3x}{x^2-4}$$ is a rational function in reduced form, because the zeros of the numerator ($-1$, $0$ and $1$) and denominator ($-2$ and $2$) are distinct. The function has asymptotes $x=-2$, $x=2$ and $y=x$.