# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/function-modeling/rational-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/function-modeling/rational-function/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Rational Function",
  description: "Master rational functions with real-world examples, domain finding, simplification techniques, and operations. Learn through practical problems like chicken coop design.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/18/2025",
  subject: "Functions and Their Modeling",
};

## 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:

<BlockMath math="f(x) = \frac{P(x)}{Q(x)}" />

Where:

- <InlineMath math="P(x)" /> is a polynomial in the numerator
- <InlineMath math="Q(x)" /> is a polynomial in the denominator
- <InlineMath math="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 <InlineMath math="x" /> meters, then:

- Area = length × width = 100
- <InlineMath math="x \times \text{width} = 100" />
- Width = <InlineMath math="\frac{100}{x}" />

The function <InlineMath math="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:

<BlockMath math="f(x) = \frac{k}{x}" />

Where <InlineMath math="k" /> is a constant. Example: <InlineMath math="f(x) = \frac{5}{x}" />

### Linear Rational Function

Both numerator and denominator are linear functions:

<BlockMath math="f(x) = \frac{ax + b}{cx + d}" />

Example: <InlineMath math="f(x) = \frac{2x + 3}{x - 1}" />

### Quadratic Rational Function

Involves quadratic functions in the numerator or denominator:

<BlockMath math="f(x) = \frac{x^2 + 2x + 1}{x - 3}" />

## Domain of Rational Functions

The domain of a rational function is all values of <InlineMath math="x" /> that make the function defined. Remember, the denominator cannot be zero!

**How to find the domain:**

1. Find values of <InlineMath math="x" /> that make the denominator = 0
2. The domain is all real numbers except those values

**Example:** Determine the domain of <InlineMath math="f(x) = \frac{x + 2}{x - 3}" />

**Solution:**

- Denominator is zero when: <InlineMath math="x - 3 = 0" />
- So: <InlineMath math="x = 3" />
- Domain: <InlineMath math="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: <InlineMath math="f(x) = \frac{6x^2}{3x}" />

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="f(x) = \frac{6x^2}{3x}" />
  <BlockMath math="= \frac{6x \cdot x}{3x}" />
  <BlockMath math="= \frac{6x}{3} \cdot \frac{x}{x}" />
  <BlockMath math="= 2x" />
</div>

### With Factoring

Simplify: <InlineMath math="f(x) = \frac{x^2 - 4}{x - 2}" />

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="f(x) = \frac{x^2 - 4}{x - 2}" />
  <BlockMath math="= \frac{(x + 2)(x - 2)}{x - 2}" />
  <BlockMath math="= x + 2" />
</div>

Note: <InlineMath math="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:** <InlineMath math="\frac{2}{x} + \frac{3}{x + 1}" />

**Solution:**

<div className="flex flex-col gap-4">
  <BlockMath math="\frac{2}{x} + \frac{3}{x + 1}" />
  <BlockMath math="= \frac{2(x + 1)}{x(x + 1)} + \frac{3x}{x(x + 1)}" />
  <BlockMath math="= \frac{2(x + 1) + 3x}{x(x + 1)}" />
  <BlockMath math="= \frac{2x + 2 + 3x}{x(x + 1)}" />
  <BlockMath math="= \frac{5x + 2}{x(x + 1)}" />
</div>

### Multiplication

Multiply numerator with numerator, denominator with denominator:

<BlockMath math="\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}" />

**Example:** <InlineMath math="\frac{x + 1}{x} \times \frac{2x}{x - 1}" />

**Solution:**

<BlockMath math="\frac{x + 1}{x} \times \frac{2x}{x - 1} = \frac{2x(x + 1)}{x(x - 1)} = \frac{2(x + 1)}{x - 1}" />

### Division

Remember, dividing = multiplying by the reciprocal:

<BlockMath math="\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}" />

## Exercises

1. Determine the domain of <InlineMath math="f(x) = \frac{x + 3}{x^2 - 9}" />

2. Simplify <InlineMath math="f(x) = \frac{x^2 - 1}{x + 1}" />

3. Calculate <InlineMath math="\frac{1}{x - 1} - \frac{2}{x + 1}" />

4. A car travels 300 km. If the average speed is <InlineMath math="v" /> km/h, write the travel time function in terms of <InlineMath math="v" />.

### Answer Key

**Answer 1:**

<div className="flex flex-col gap-4">
  <BlockMath math="x^2 - 9 = 0" />
  <BlockMath math="(x + 3)(x - 3) = 0" />
  <BlockMath math="x = -3 \text{ or } x = 3" />
</div>

Domain: <InlineMath math="D_f = \{x | x \neq -3, x \neq 3, x \in \mathbb{R}\}" />

**Answer 2:**

<div className="flex flex-col gap-4">
  <BlockMath math="f(x) = \frac{x^2 - 1}{x + 1}" />
  <BlockMath math="= \frac{(x + 1)(x - 1)}{x + 1}" />
  <BlockMath math="= x - 1" />
</div>

With the condition <InlineMath math="x \neq -1" />

**Answer 3:**

<div className="flex flex-col gap-4">
  <BlockMath math="\frac{1}{x - 1} - \frac{2}{x + 1}" />
  <BlockMath math="= \frac{(x + 1) - 2(x - 1)}{(x - 1)(x + 1)}" />
  <BlockMath math="= \frac{x + 1 - 2x + 2}{(x - 1)(x + 1)}" />
  <BlockMath math="= \frac{-x + 3}{(x - 1)(x + 1)}" />
</div>

**Answer 4:**

Time = Distance ÷ Speed

<BlockMath math="t(v) = \frac{300}{v}" />
