FUNCTIONS 07
Rational Functions
A rational function is a quotient of polynomial functions. Its denominator can create excluded inputs, asymptotes and discontinuities.
The main idea
A rational function has the form
\[\boxed{f(x)=\frac{P(x)}{Q(x)},\qquad Q(x)\ne0}.\]The condition \(Q(x)\ne0\) is essential because division by zero is undefined.
1. The reciprocal function
The simplest rational function is
\[f(x)=\frac1x.\]Its domain excludes \(x=0\). Positive inputs give positive outputs and negative inputs give negative outputs.
2. Vertical asymptotes
For
\[f(x)=\frac{x+1}{x-3},\]the denominator is zero at \(x=3\). Near this value the function becomes very large in magnitude, so \(x=3\) is a vertical asymptote.
3. Horizontal asymptotes
For \(f(x)=1/x\),
\[\frac1x\to0\quad\text{as}\quad |x|\to\infty.\]The graph approaches \(y=0\) but never reaches it.
For \((x+1)/(x-3)\), the highest powers have the same degree, so the graph approaches the ratio of leading coefficients:
\[y=1.\]horizontal asymptote: \(y=0\)
\((x+1)/(x-3)\to1\)
horizontal asymptote: \(y=1\)
4. Intercepts still come from the formula
A rational function is zero when its numerator is zero, provided the denominator is not zero there.
For
\[f(x)=\frac{x+1}{x-3},\]the \(x\)-intercept is \((-1,0)\), because \(x+1=0\) at \(x=-1\).
The \(y\)-intercept is
\[f(0)=-\frac13.\]5. Cancelling factors does not restore an excluded input
Consider
\[f(x)=\frac{x^2-1}{x-1}=\frac{(x-1)(x+1)}{x-1}.\]For \(x\ne1\), this simplifies to \(x+1\). But the original denominator is zero at \(x=1\), so that input remains excluded.
↓ factorise
\(\frac{(x-1)(x+1)}{x-1}\)
↓ cancel only for \(x\ne1\)
\(y=x+1\), with \(x=1\) still missing
6. A removable discontinuity is a hole
The simplified expression behaves exactly like the line \(y=x+1\), except at \(x=1\).
The missing point would be \((1,2)\), so the graph has an open circle there.
7. Vertical asymptote and hole are different
A vertical asymptote occurs when function values grow without bound near an excluded input.
A hole occurs when the problematic factor cancels and the nearby values approach one finite point.
\(x\to a\Rightarrow |f(x)|\to\infty\)
removable hole
\(x\to a\Rightarrow f(x)\to L\), finite
8. Limits describe behaviour near excluded points
Although
\[\frac{x^2-1}{x-1}\]is undefined at \(x=1\), nearby values follow \(x+1\). Therefore
\[\boxed{\lim_{x\to1}\frac{x^2-1}{x-1}=2}.\]The limit describes what the function approaches, not necessarily what it equals at the point.
↓
nearby function values
\(\to2\)
even though \(f(1)\) is undefined
9. General end behaviour
For \(P(x)/Q(x)\), compare the degrees of numerator and denominator.
If \(\deg P<\deg Q\), the horizontal asymptote is usually \(y=0\). If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If \(\deg P=\deg Q+1\), a slant asymptote may occur.
same degree → ratio of leading coefficients
numerator one degree higher → possible slant asymptote
10. Why rational functions matter for calculus
Rational functions naturally bring together domain restrictions, limits, continuity, asymptotes and derivatives.
\[\text{rational functions}\to\text{limits}\to\text{continuity}\to\text{derivatives}.\]↓ domain restriction
graph near excluded value
↓ limit
continuity and derivative questions
Rational functions are quotients of polynomials. Denominator zeros must be excluded. Some exclusions produce vertical asymptotes, while cancelled factors produce holes. Horizontal behaviour is controlled by the highest-degree terms.