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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 $$y=x$$.