Supercharge your learning!

Use adaptive quiz-based learning to study this topic faster and more effectively.

Examples of rational functions

Examples of rational functions are $$$ f(x) = \frac{ax+b}{dx+e},\quad g(x) = \frac{ax^2+bx+c}{dx+e}.$$$ We assume that $$a,d\ne 0$$ and that $$-e/d$$ is not a zero of the numerator, so that the functions are in reduced form

  • Domain. The functions are defined on $$\R\setminus \{-e/d\}$$
  • Zero. Their zeros are the zeros of the numerator.
  • Vertical asymptote: $$x=-e/d$$.
  • Horizontal asymptote for $$f$$: $$\displaystyle y = a/d$$.
  • Oblique asymptote for $$g$$. $$$\displaystyle y = \alpha x + \beta,\quad \alpha = \frac{a}{d},\; \beta=\frac{bd-ae}{d^2}.$$$

The oblique asymptote for $$g$$ is found from the computations ($$x\to\infty$$) \begin{align*} g(x) - \alpha x-\beta &= \frac{ ax^2+bx+c -\alpha d x^2 - (\alpha e + \beta d)x - \beta e}{ dx+e }\\ &= \frac{cd^2-bde+ae^2}{d^2(dx+e)}\to 0. \end{align*}

Rational functions: $$f(x)=\frac{x}{x+1}$$ and $$g(x)=\frac{x^2-1}{x}$$
Rational functions: $$f(x)=\frac{x}{x+1}$$ and $$g(x)=\frac{x^2-1}{x}$$