Rational Functions

Definition:

A rational function is any function that can be expressed as the ratio of two polynomials, where the function takes the form:

\[ f(x) = \frac{P(x)}{Q(x)} \]

where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \neq 0 \) because division by zero is undefined. The domain of a rational function includes all real values of 𝑥 for which 𝑄(𝑥) ≠ 0.

Example 1: Simple Rational Function

Let’s start with a simple rational function: $$ f(x) = \frac{1}{x} $$ Here, 𝑃(𝑥) = 1 (a constant polynomial) and 𝑄(𝑥) = 𝑥. To find the domain, we identify the values of 𝑥 that make 𝑄(𝑥) = 0. In this case, 𝑥 = 0 would make the denominator zero, so we exclude it from the domain.

Domain: All real numbers except 𝑥 = 0, or in interval notation, (−∞ ,0)∪(0,∞).

Example 2: Rational Function with a Quadratic Denominator

Consider the function: $$ f(x) = \frac{x + 2}{x^2 - 4} $$ Here, \(𝑃(𝑥) = 𝑥 + 2 \) and \( 𝑄(𝑥) = 𝑥^2 − 4\).
We factor the denominator as $$ 𝑄( 𝑥 ) = ( 𝑥 − 2 ) ( 𝑥 + 2) $$ which shows us that the values 𝑥 = 2 and 𝑥 = − 2 would make 𝑄 ( 𝑥 ) = 0.

Domain: All real numbers except 𝑥 = ±2, or in interval notation, $$ (− ∞ , − 2 ) ∪ ( − 2 , 2) ∪ (2 , ∞). $$

Example 3: Rational Function with a Higher-Degree Polynomial Denominator

Consider the function: $$ f(x) = \frac{x^2 + x -2}{x^3 - x} $$ Here, we have $$ P(x) = x^2 + x -2 $$ and $$ Q(x) = x^3 - x $$ Factoring the denominator: $$ P(x) = x (x - 1) (x + 1) $$ The denominator is zero when $$ x = 0, \quad x = - 1, \quad or \quad x = 1 $$

Domain: All real numbers except \( x = -1 \) , \( x = 0 \) , or \( x = 1 \) or in interval notation, $$ (- \infty, -1) U (-1, 1) U (1, \infty) $$

Example 4: \[ f(x) = \frac{3x}{x + 1} \]

In this example, \( f(x) \) has a polynomial in the numerator and the denominator, with a domain of all real numbers except \( x = -1 \) where the denominator is zero. $$ (- \infty, - 1) U (- 1, \infty) $$

Example 5: \[ f(x) = \frac{x^2 + 5x + 6}{x + 2} \]

The domain here is all real numbers except \( x = -2 \) because that value makes the denominator zero. $$ (- \infty, - 2) U (- 2, \infty) $$

Example 6: \[ f(x) = \frac{(x-3)(x+2)}{(x-3)(x-1)} \]

Applications of Rational Functions

• Electrical Resistance: \[ R = \frac{R_1 R_2}{R_1 + R_2} \]
• Drug Concentration: \[ C(t) = \frac{d}{V + kt} \]
• Average Cost: \[ A(x) = \frac{C(x)}{x} \]
• Gravitational Force: \[ F = \frac{Gm_1m_2}{r^2} \]

Key Points:

• Rational functions are undefined where the denominator is zero.
• Domain is found by excluding values that make the denominator zero.
• Simplifying a rational function does not change the domain, even if terms cancel in the simplified form.

Quadratic Denominator with Complex Roots

To have a rational function where the domain is all real numbers, we need the denominator to never be zero for any real values of 𝑥. This can happen if the denominator is a polynomial with no real roots—typically, a quadratic (or higher-degree) polynomial that has complex roots only. Here are some examples of such rational functions:

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

• Numerator: \(P(x) = x + 1 \)
• Denominator: \(Q(x) = x^2 + 1 \)

To determine if there are any real roots in the denominator, we set $$ x^2 + 1 = 0: $$ $$ x^2 = - 1 $$ This equation has no real solutions since \(x = \pm i\) (where 𝑖 i is the imaginary unit). Therefore, \( Q(x) \ne 0\) for any real 𝑥, and the domain of 𝑓(𝑥) is all real numbers.

Domain: All real numbers, or \( (- \infty, \infty) \)

Example 2: $$ f(x) = \frac{2x + 3}{x^4 + 4} $$ Here, To check if \( Q(x) = 0\), has any real roots, we solve: $$ x^4 + 4 = 0: $$ $$ x^4 = - 4 $$ Again, this equation has no real solutions, as \(x^4 \) cannot be negative for any real 𝑥. Thus \( Q(x) \ne 0\) for all real 𝑥, making the domain all real numbers.

Domain: All real numbers, or \( (- \infty, \infty) \)

Example 3: $$ f(x) = \frac{3x - 5}{x^2 - 2x + 2} $$ Here,

• Numerator: \(P(x) = 3x - 5 \)
• Denominator: \(Q(x) = x^2 - 2x + 2 \)

To find the roots of \(Q(x) = 0 \), we use the quadratic formula: $$ x = \frac{2 \pm \sqrt((-2)^2 - 4.1.2)}{2.1} $$ $$ x = \frac{2 \pm \sqrt(4 - 8)}{2} $$ $$ x = \frac{2 \pm \sqrt(-4)}{2} $$ $$ x = 2 \pm 2 i $$ $$ x = 1 \pm i $$ The roots $$ x = 2 \pm 2 i $$ are complex, so \( Q(x) \ne 0 \) for all real 𝑥.

Domain: All real numbers, or \( (- \infty, \infty) \)

Summary

In each example, the denominator has no real roots, only complex roots, meaning there are no values of 𝑥 that would make the denominator zero. As a result, the domain of each of these rational functions is all real numbers , \( (- \infty, \infty) \).