Rational Functions, Holes, and Asymptotes

Definition A rational function is a function of the form

where $p$ and $q$ are polynomials and $q$ is not the zero polynomial. Unless a domain is stated, the domain of $f$ is the set of all real numbers $x$ with $q(x) \ne 0$.

Definition The excluded inputs of a rational function $p(x)/q(x)$ are the real zeros of the original denominator $q$. They must be found before any factor is cancelled. If $p(x) = c(x)\,P(x)$ and $q(x) = c(x)\,Q(x)$, then

so the simplified formula agrees with $f$ on the domain of $f$ but does not enlarge it.

Definition The reciprocal graph is the graph of $y = \frac{1}{x}$, defined for $x \ne 0$. It has the vertical asymptote $x = 0$ and the horizontal asymptote $y = 0$, and its two branches lie in the first and third quadrants. The graph of

has the same shape, moved so that its asymptotes are $x = h$ and $y = k$, scaled vertically by the factor $\lvert a \rvert$ and, when $a < 0$, reflected in the line $y = k$, which puts the branches to the upper left and lower right of the point $(h, k)$ where the asymptotes cross.

Definition Let $f = p/q$. Cancel every common factor of $p$ and $q$ to write $f$ in lowest terms as $P/Q$, so that $P$ and $Q$ share no factor of positive degree and $f(x) = P(x)/Q(x)$ for every $x$ in the domain of $f$. If $a$ is a zero of the original denominator $q$ but $Q(a) \ne 0$, then the graph of $f$ has a hole at the point

The point is missing from the graph because $a$ is not in the domain, while nearby outputs come as close to $P(a)/Q(a)$ as you like.

Definition The line $x = a$ is a vertical asymptote of $f$ if $f(x)$ grows without bound in size as $x$ approaches $a$ from at least one side, that is, $f(x) \to \infty$ or $f(x) \to -\infty$ as $x \to a^{+}$ or as $x \to a^{-}$. For a rational function written in lowest terms as $P/Q$, as in the previous section, the vertical asymptotes are exactly the lines $x = a$ with $Q(a) = 0$. At such an $a$ the numerator satisfies $P(a) \ne 0$, because $P$ and $Q$ share no factor, so near $a$ the output is roughly a fixed nonzero number divided by a tiny one. A zero $a$ of $q$ survives into $Q$ exactly when the factor $x - a$ appears more times in $q$ than in $p$, and then it gives a vertical asymptote, not a hole.

Definition The line $y = L$ is a horizontal asymptote of $f$ if $f(x) \to L$ as $x \to \infty$ or as $x \to -\infty$. For $f = p/q$ with $p$ not the zero polynomial, $\deg p = n$, $\deg q = m$ and leading coefficients $a_n$ and $b_m$,

If $n > m$, there is no horizontal asymptote. For a rational function the same line serves at both ends. A horizontal asymptote describes only the far ends of the graph. The graph may cross it at finite inputs.

Definition The domain of a rational function is a union of open intervals whose endpoints are the excluded inputs. On each of these intervals the graph is one unbroken curve, with no gaps or jumps, and the part of the graph over one such interval is called a branch. Across a vertical asymptote neighboring branches are genuinely separate, since for a rational function the outputs grow without bound in size on both sides of the asymptote. Across a hole they line up, and only the single point is missing. By this definition a line with a hole has two branches that meet at the missing point, so they look like one line with a gap.

Definition The $y$-intercept of $f$ is the point $(0, f(0))$, which exists only when $0$ is in the domain. The $x$-intercepts are the points $(a, 0)$ with $a$ in the domain and $f(a) = 0$. For $f = p/q$ these are the zeros of the numerator $p$ that are not excluded inputs.