# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/exponential-logarithm/function-exploration
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/exponential-logarithm/function-exploration/en.mdx

Output docs content for large language models.

---

import { VirusChart } from "./virus-chart";

export const metadata = {
   title: "Function Exploration",
   description: "Explore exponential functions through interactive virus spread visualization. Compare exponential vs linear growth with real-time charts.",
   authors: [{ name: "Nabil Akbarazzima Fatih" }],
   date: "04/02/2025",
   subject: "Exponents and Logarithms",
};

## Introduction to Exponential Functions

Exponential functions are mathematical functions that can describe extremely rapid growth or decay. Let's explore the properties of exponential functions through a real-world example.

## Virus Spread

Imagine the following scenario: Someone carries a virus and infects 3 other people. Then, each of those people infects 3 more people in the next phase.

### Spread Pattern

If we track the number of infected people in each phase:

- **Phase 1**: <InlineMath math="3 = 3^1"/> people infected
- **Phase 2**: <InlineMath math="9 = 3^2"/> people infected
- **Phase 3**: <InlineMath math="27 = 3^3"/> people infected
- **Phase 4**: <InlineMath math="81 = 3^4"/> people infected
- **Phase 5**: <InlineMath math="243 = 3^5"/> people infected

### Mathematical Pattern

From the data above, a clear pattern emerges: the number of people infected in phase <InlineMath math="x"/> is <InlineMath math="3^x"/>.

If <InlineMath math="f(x)"/> represents the number of people infected in phase <InlineMath math="x"/>, then:

<BlockMath math="f(x) = 3^x" />

This is an example of an exponential function.

### Spread Visualization

1. **Exponential Graph**: <InlineMath math="f(x) = 3^x"/>
2. **Linear Graph**: <InlineMath math="f(x) = 3x"/>
3. **Logarithmic Graph**: <InlineMath math="f(x) = \log(x+1) \cdot 20"/>

Using the equations above, we can visualize the virus spread through the following graph:

<VirusChart />

### Analysis Questions

1. How many people will be infected in phase 20?

   <BlockMath math="f(20) = 3^{20} = 3,486,784,401" />

2. Which function best represents the virus spread?

   Of the three graphs shown, the **exponential graph** most accurately depicts this virus spread. This graph shows slow growth initially but becomes very rapid as phases progress.

## Properties of Exponential Functions

From the exploration above, we can conclude several properties of the exponential function <InlineMath math="f(x) = a^x"/> (where <InlineMath math="a > 0"/> and <InlineMath math="a \neq 1"/>):

1. **Rapid Growth/Decay**: Function values increase/decrease very rapidly.
2. **Domain and Range**: Domain is all real numbers, range is all positive numbers.
3. **Intercept Point**: Always passes through point (0,1) because <InlineMath math="a^0 = 1"/>.
4. **Graph Properties**:
   - If <InlineMath math="a > 1"/>, the function increases (as in the virus spread case with <InlineMath math="a = 3"/>)
   - If <InlineMath math="0 < a < 1"/>, the function decreases

## Applications of Exponential Functions

Exponential functions are used in various fields:

- Population growth
- Compound interest in economics
- Radioactive decay
- Disease spread (as in the example above)

Understanding exponential functions helps us analyze and predict phenomena that exhibit rapid growth or decay.