Rational Function

What is a Rational Function?

Have you ever seen fractions in mathematics? Well, rational functions are similar to fractions, but more interesting because they involve variables!

A rational function is a function in the form of a fraction, where both the numerator and denominator are polynomial functions. Simply put, a rational function can be written as:

Where:

• $P(x)$ is a polynomial in the numerator
• $Q(x)$ is a polynomial in the denominator
• $Q(x) \neq 0$ (denominator cannot be zero)

Examples of Rational Functions in Life

Let's look at a real example to better understand rational functions.

Chicken Coop Problem:

Mr. Budi wants to build a rectangular chicken coop with an area of 100 m². He wants to know the relationship between the length and width of the coop.

If the length of the coop is $x$ meters, then:

• Area = length × width = 100
• $x \times \text{width} = 100$
• Width = $\frac{100}{x}$

The function $f(x) = \frac{100}{x}$ is an example of a rational function!

Types of Rational Functions

Simple Rational Function

The simplest form of a rational function:

Where $k$ is a constant. Example: $f(x) = \frac{5}{x}$

Linear Rational Function

Both numerator and denominator are linear functions:

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

Quadratic Rational Function

Involves quadratic functions in the numerator or denominator:

Domain of Rational Functions

The domain of a rational function is all values of $x$ that make the function defined. Remember, the denominator cannot be zero!

How to find the domain:

1. Find values of $x$ that make the denominator = 0
2. The domain is all real numbers except those values

Example: Determine the domain of $f(x) = \frac{x + 2}{x - 3}$

Solution:

• Denominator is zero when: $x - 3 = 0$
• So: $x = 3$
• Domain: $D_f = \{x | x \neq 3, x \in \mathbb{R}\}$

Simplifying Rational Functions

Rational functions can be simplified by finding common factors in the numerator and denominator.

Without Factoring

Simplify: $f(x) = \frac{6x^2}{3x}$

Solution:

With Factoring

Simplify: $f(x) = \frac{x^2 - 4}{x - 2}$

Solution:

Note: $x \neq 2$ (from the original domain)

Operations on Rational Functions

Addition and Subtraction

Just like regular fractions, we need to find a common denominator first!

Example: $\frac{2}{x} + \frac{3}{x + 1}$

Solution:

Multiplication

Multiply numerator with numerator, denominator with denominator:

Example: $\frac{x + 1}{x} \times \frac{2x}{x - 1}$

Solution:

Division

Remember, dividing = multiplying by the reciprocal:

Exercises

1. Determine the domain of $f(x) = \frac{x + 3}{x^2 - 9}$

2. Simplify $f(x) = \frac{x^2 - 1}{x + 1}$

3. Calculate $\frac{1}{x - 1} - \frac{2}{x + 1}$

4. A car travels 300 km. If the average speed is $v$ km/h, write the travel time function in terms of $v$.

Answer Key

Answer 1:

Domain: $D_f = \{x | x \neq -3, x \neq 3, x \in \mathbb{R}\}$

Answer 2:

With the condition $x \neq -1$

Answer 3:

Answer 4:

Time = Distance ÷ Speed