Rational Function
A rational function is a function that can be expressed as the ratio of two polynomials, f(x) = P(x)/Q(x), where Q(x) ≠ 0. Its domain excludes all values of x that make the denominator zero, and its graph exhibits vertical asymptotes at these excluded values and may have horizontal or oblique asymptotes determined by the degrees of P and Q. Rational functions model phenomena such as average cost, concentration over time in pharmacokinetics, and electrical impedance in circuits.
Key Formula
f(x) = P(x)/Q(x), with degree m and degree n
LaTeX: f(x) = \frac{P(x)}{Q(x)},\quad \deg(P) = m,\; \deg(Q) = n
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | rational function value | dimensionless |
| P(x) | numerator polynomial of degree m | dimensionless |
| Q(x) | denominator polynomial of degree n, Q(x)≠0 | dimensionless |
| m, n | degrees of numerator and denominator respectively | dimensionless |
Worked Example
Problem
Find the domain, vertical asymptotes, and horizontal asymptote of f(x) = (2x² + 1)/(x² − 4).
Solution
Step 1 (Domain): Set denominator ≠ 0: x² − 4 ≠ 0 → x ≠ ±2. Domain: (−∞,−2)∪(−2,2)∪(2,∞). Step 2 (Vertical Asymptotes): x = 2 and x = −2 (denominator is 0 there, numerator ≠ 0). Step 3 (Horizontal Asymptote): Both P and Q have degree 2. Leading coefficients: 2 and 1. Horizontal asymptote: y = 2/1 = 2. Step 4: The function approaches y = 2 as x → ±∞.
Answer
Domain: x ≠ ±2; Vertical asymptotes: x = 2 and x = −2; Horizontal asymptote: y = 2
Horizontal Asymptote Rules for Rational Functions f(x) = P(x)/Q(x)
| Condition | Relationship | Horizontal Asymptote | Example |
|---|---|---|---|
| deg P < deg Q | m < n | y = 0 | 1/(x²+1) → y=0 |
| deg P = deg Q | m = n | y = leading coeff ratio | (2x)/(3x) → y=2/3 |
| deg P > deg Q | m > n | None (oblique or none) | (x²)/(x+1) → oblique |
| deg P = deg Q + 1 | m = n+1 | Oblique asymptote | (x²+1)/x → y=x |
Interactive Tools
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Related Terms
Rational Expression
A rational expression is a fraction in which both the numerator and denominator are polynomials, and the denominator is not zero. It represents the ratio of two polynomial functions and is defined for all values of the variable except those that make the denominator equal to zero. Rational expressions are used in modeling real-world phenomena, simplifying algebraic fractions, and solving equations in calculus and engineering.
Partial Fractions
Partial fraction decomposition is the process of expressing a rational expression as a sum of simpler fractions whose denominators are the factors of the original denominator. This technique reverses the process of adding fractions and is essential for integrating rational functions in calculus, solving differential equations, and performing inverse Laplace transforms in engineering. Each simpler fraction is called a partial fraction.
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
The term "rational function" mirrors "rational number" in that both are expressed as ratios: a rational number is a ratio of integers, and a rational function is a ratio of polynomials. The word "rational" derives from the Latin "ratio," meaning "reckoning" or "ratio." Systematic study of rational functions developed alongside calculus in the 17th century.