Rational Function

A rational function is a quotient of polynomial functions and plays an important role in calculus. Understanding the continuity properties of rational functions is essential for studying more complicated types of function.

Basic Properties

A rational function has the following basic properties:

• Domain: all real numbers except the points where the denominator is zero
• Continuity: continuous on its domain
• Points of discontinuity: discontinuous at the points where the denominator is zero
• Asymptotes: may have a vertical asymptote and a horizontal asymptote

Definition of Rational Functions

Common Rational Functions

1. Simple Rational Functions

Example 1: $f(x) = \frac{1}{x}$

Analysis:

• Domain: $x \neq 0$
• Discontinuous at $x = 0$
• Has a vertical asymptote $x = 0$
• Has a horizontal asymptote $y = 0$

Example 2: $f(x) = \frac{x^2 - 1}{x - 1}$

Analysis:

• Domain: $x \neq 1$
• Discontinuous at $x = 1$
• Can be simplified to $f(x) = x + 1$ (when $x \neq 1$)
• If we define $f(1) = 2$, then the function is continuous at $x = 1$

2. Complicated Rational Functions

Example 3: $f(x) = \frac{1}{x^2 + 1}$

Analysis:

• Domain: $\mathbb{R}$ (the denominator is always greater than zero)
• Continuous everywhere on $\mathbb{R}$
• The graph is a smooth curve
• Has a horizontal asymptote $y = 0$

Example 4: $f(x) = \frac{x^2 + 1}{x}$

Analysis:

• Domain: $x \neq 0$
• Discontinuous at $x = 0$
• Has a vertical asymptote $x = 0$
• Has no horizontal asymptote

Continuity Analysis of Rational Functions

Determining Continuity

The continuity of a rational function $f(x) = \frac{P(x)}{Q(x)}$:

1. Continuous on its domain: continuous at every point where $Q(x) \neq 0$
2. Points of discontinuity: discontinuous at every point where $Q(x) = 0$
3. Removable discontinuities: if $P(x_0) = Q(x_0) = 0$ and $\lim_{x \to x_0} \frac{P(x)}{Q(x)}$ exists, then the function can be made continuous at that point by defining $f(x_0)$

Types of Point of Discontinuity

1. Removable discontinuity: the limit exists but the function value is undefined
2. Jump discontinuity: the left and right limits exist but are not equal
3. Infinite discontinuity: the limit tends to infinity

Graph Characteristics of Rational Functions

Summary of Graph Features

1. Smoothness: on its domain, the graph of a rational function is a smooth curve
2. Asymptotes: may have a vertical asymptote and a horizontal asymptote
3. Points of discontinuity: at the points where the denominator is zero there is a vertical asymptote or a removable discontinuity
4. Local behaviour: near a point of continuity, the function values change gradually

Applications of Rational Functions

1. Function Modelling

Rational functions are widely used in science and engineering:

• Physical modelling: describe various physical phenomena
• Economic modelling: describe economic relationships such as costs and revenues
• Engineering applications: describe various engineering parameters

2. Function Approximation

Rational functions can be used to approximate other complicated functions:

• Padé approximation: approximate a complicated function with rational functions
• Interpolation: construct a rational function through given points

Exercises

Exercise 1

Determine the continuity of $f(x) = \frac{x^2 - 4}{x - 2}$ at $x = 2$.

Exercise 2

Determine the continuity of $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.

Exercise 3

Let $f(x) = \begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1 \\ a, & x = 1 \end{cases}$. Find the value of the constant $a$ that makes $f(x)$ continuous at $x = 1$.

Exercise 4

Let $f(x) = \frac{x^2 - 1}{x - 1}$. Find the intervals of continuity of $f(x)$.

Exercise 5

Determine the continuity of $f(x) = \frac{x^3 + 1}{x^2 - 1}$ at $x = 1$.

Exercise 6

Let $f(x) = \begin{cases} \frac{x^3 - 1}{x - 1}, & x \neq 1 \\ a, & x = 1 \end{cases}$. Find the value of the constant $a$ that makes $f(x)$ continuous at $x = 1$.

---

Summary

Symbols Used in This Article