Rational Functions

Definition

A rational function is the ratio of two polynomials:

f(x) = p(x) / q(x)

where p and q are polynomials, and q is not the zero polynomial. The function is defined wherever q(x) ≠ 0.

Examples:

• 1/x (simplest non-trivial rational function).
• (x² + 1)/(x − 2).
• (x³ − 3x + 2)/(x² + 1).

Asymptotes — vertical and horizontal

Vertical asymptotes

At x = a where q(a) = 0 and p(a) ≠ 0, the function has a vertical asymptote. f(x) → ±∞ as x → a (sign depending on direction and sign of p(a)).

For f(x) = 1/(x − 3), there's a vertical asymptote at x = 3.

Horizontal asymptotes

The behavior as x → ±∞ depends on the relative degrees of p and q:

Comparison	Horizontal asymptote
deg p < deg q	y = 0
deg p = deg q	y = a_p / a_q (leading coefficient ratio)
deg p = deg q + 1	Oblique (slant) asymptote y = mx + b (perform polynomial division)
deg p > deg q + 1	None — function grows without bound

For (3x² + 5)/(2x² − 1), horizontal asymptote y = 3/2.

Holes — removable discontinuities

If p(x) and q(x) share a common factor (x − a), the function has a hole at x = a — undefined there, but cancellation gives a finite limit.

Example — f(x) = (x² − 1) / (x − 1) = (x − 1)(x + 1) / (x − 1) = x + 1 for x ≠ 1.

The graph looks like the line y = x + 1, but with a missing point at (1, 2). The discontinuity is "removable" — by defining f(1) = 2, the function becomes continuous.

Partial fractions decomposition

Any proper rational function (deg p < deg q) can be written as a sum of simpler rational functions. The simpler pieces are determined by the factorization of q.

For q(x) factored into distinct linear factors:

p(x) / [(x − r₁)(x − r₂)(x − r₃)] = A₁/(x − r₁) + A₂/(x − r₂) + A₃/(x − r₃)

For repeated factors:

p(x) / (x − r)² = A/(x − r) + B/(x − r)²

For irreducible quadratic factors:

p(x) / [(x − r)(x² + bx + c)] = A/(x − r) + (Bx + C)/(x² + bx + c)

Solve A, B, C, ... by matching coefficients or substituting values.

Integrating rational functions

Partial fractions reduces integration of rational functions to integrating simpler pieces, all of which have closed forms:

Form	Integral
1 / (x − a)	ln|x − a| + C
1 / (x − a)²	−1 / (x − a) + C
1 / (x² + a²)	(1/a) arctan(x/a) + C
x / (x² + a²)	(1/2) ln(x² + a²) + C

Any rational function can be integrated using these pieces. This is one reason partial fractions is foundational.

Transfer functions in control theory

A linear time-invariant system has a transfer function:

H(s) = Y(s) / X(s) = B(s) / A(s)

where X is input and Y is output (in Laplace transform), and B and A are polynomials.

Key features:

• Poles — roots of A(s). Determine system stability — system stable iff all poles in left-half plane (negative real part).
• Zeros — roots of B(s). Don't affect stability but shape the response.
• DC gain — H(0). Steady-state output for a constant input.

Engineers analyze and design systems by manipulating the rational transfer function — adding poles/zeros via controllers, doing pole placement, etc.

JavaScript — rational function tools

// Evaluate a rational function p(x)/q(x), with polynomials as arrays
// pCoeffs = [a0, a1, a2, ...] for a0 + a1*x + a2*x² + ...
function evalPoly(coeffs, x) {
  return coeffs.reduce((acc, c, i) => acc + c * Math.pow(x, i), 0);
}

function rational(pCoeffs, qCoeffs, x) {
  return evalPoly(pCoeffs, x) / evalPoly(qCoeffs, x);
}

// Example: (x² + 1) / (x - 2) at x = 5
console.log(rational([1, 0, 1], [-2, 1], 5));  // 26/3 ≈ 8.667

// Find vertical asymptotes (zeros of q where p is nonzero)
function verticalAsymptotes(pCoeffs, qCoeffs, candidates) {
  return candidates.filter(x => {
    const qVal = evalPoly(qCoeffs, x);
    const pVal = evalPoly(pCoeffs, x);
    return Math.abs(qVal) < 1e-9 && Math.abs(pVal) > 1e-9;
  });
}

console.log(verticalAsymptotes([1, 0, 1], [-2, 1], [-1, 0, 1, 2]));  // [2]

// Horizontal asymptote based on degrees
function horizontalAsymptote(pCoeffs, qCoeffs) {
  // Strip trailing zeros to get true degrees
  const pDeg = pCoeffs.length - 1;
  const qDeg = qCoeffs.length - 1;
  if (pDeg < qDeg) return 0;
  if (pDeg === qDeg) return pCoeffs[pDeg] / qCoeffs[qDeg];
  return null;  // no horizontal asymptote
}

console.log(horizontalAsymptote([1, 0, 3], [1, 0, 2]));  // 3/2 = 1.5
console.log(horizontalAsymptote([1, 0], [1, 0, 1]));     // 0
console.log(horizontalAsymptote([1, 0, 0, 1], [1, 0, 1])); // null

// Partial fractions for (px + q) / ((x - a)(x - b))
// = A/(x - a) + B/(x - b)
function partialFractions2(p, q, a, b) {
  // A = (p*a + q) / (a - b)
  // B = (p*b + q) / (b - a)
  return {
    A: (p * a + q) / (a - b),
    B: (p * b + q) / (b - a)
  };
}

// (3x + 5) / ((x - 1)(x + 2)) = ?/(x - 1) + ?/(x + 2)
console.log(partialFractions2(3, 5, 1, -2));
// A = (3 + 5) / (1 - (-2)) = 8/3
// B = (-6 + 5) / (-3) = 1/3

Where rational functions show up

• Calculus. Integration via partial fractions. Foundation for solving ODEs with rational coefficients (Frobenius method, etc.).
• Control theory. Transfer functions are rational. PID controllers, lead-lag compensators, all manipulate rational H(s).
• Signal processing. z-transform of digital filters gives rational H(z). IIR (infinite impulse response) filters are pole-zero designs.
• Numerical analysis. Padé approximants — best rational approximation matching Taylor series. Converges faster than polynomial truncation, handles poles.
• Complex analysis. Meromorphic functions have rational local structure. Riemann sphere — rational functions are exactly meromorphic functions.
• Algebra and number theory. Rational functions over a field form a field (the field of fractions). Used in algebraic geometry, function fields.
• Physics — perturbation theory. Energy levels, scattering amplitudes often expand as rational functions of coupling constants.

Common mistakes

• Confusing holes with vertical asymptotes. Hole — both p(a) = 0 and q(a) = 0; cancellation yields finite limit. Vertical asymptote — q(a) = 0 but p(a) ≠ 0; function blows up. Always check both.
• Forgetting to check for common factors first. Always factor and simplify before declaring a "vertical asymptote." (x − 1)/(x − 1) is the constant 1 with a hole at x = 1, not a vertical asymptote.
• Wrong horizontal asymptote when degrees match. deg p = deg q gives y = ratio of LEADING coefficients (highest-degree). Not the constant terms or random others.
• Missing oblique asymptotes. If deg p = deg q + 1, the function approaches a line (not a constant) at infinity. Polynomial-divide to find it.
• Partial fractions with wrong template. For repeated factors (x − a)^k — need terms A/(x−a), B/(x−a)², ..., up to degree k. For irreducible quadratics — need (Bx+C)/(quadratic). Wrong template gives unsolvable system.
• Treating rational functions as just "fractions." They're algebraically richer — they form a field, support partial fractions, have meromorphic structure in ℂ. The "fraction" view misses the geometry.