# Nakafa Framework: LLM

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

Output docs content for large language models.

---

import { FunctionChart } from "@repo/design-system/components/contents/function-chart";

export const metadata = {
  title: "Function Definition",
  description: "Master exponential function definition f(x)=a^x with domain, range & properties. Understand base conditions and real-world applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/02/2025",
  subject: "Exponents and Logarithms",
};

## Exponential Function

An exponential function is a function expressed in the form:

<BlockMath math="f(x) = n \times a^x" />

with the conditions:

- <InlineMath math="a" /> is the base number, where <InlineMath math="a > 0" /> and <InlineMath math="a \neq 1" />
- <InlineMath math="n" /> is a non-zero real number
- <InlineMath math="x" /> is any real number

Exponential functions have a special characteristic where the variable <InlineMath math="x" /> is in the exponent position. This is what distinguishes exponential functions from ordinary algebraic functions. In exponential functions, small changes in the value of <InlineMath math="x" /> can result in very large changes in the function's output.

## Properties of Exponential Functions

The exponential function <InlineMath math="f(x) = a^x" /> (for <InlineMath math="n = 1" />) has several important properties:

1. The domain of the function is all real numbers (<InlineMath math="\mathbb{R}" />)
2. The range of the function is all positive numbers (<InlineMath math="\mathbb{R}^+" />)
3. It intersects the Y-axis at point <InlineMath math="(0, 1)" /> because <InlineMath math="a^0 = 1" />
4. The function is always positive for all values of x because <InlineMath math="a > 0" />
5. If <InlineMath math="a > 1" />, the function increases (monotonically increasing)
6. If <InlineMath math="0 < a < 1" />, the function decreases (monotonically decreasing)

## Special Cases of Exponential Functions

### When a = 1

If <InlineMath math="a = 1" />, then:

<BlockMath math="f(x) = n \times 1^x = n" />

The value of <InlineMath math="1^x" /> is always 1 for any value of <InlineMath math="x" />. As a result, the function becomes a constant function <InlineMath math="f(x) = n" />, no longer an exponential function. Its graph will be a horizontal line intersecting the Y-axis at point <InlineMath math="(0, n)" />.

<FunctionChart
  a={1}
  p={1}
  title="Constant Function"
  description={
    <>
      Line is always horizontal constant at <InlineMath math="y = 1" />.
    </>
  }
/>

### When a = 0

If <InlineMath math="a = 0" />, then:

<BlockMath math="f(x) = n \times 0^x" />

- For <InlineMath math="x > 0" />, the value of <InlineMath math="0^x = 0" /> so <InlineMath math="f(x) = 0" />
- For <InlineMath math="x = 0" />, the value of <InlineMath math="0^0" /> is undefined
- For <InlineMath math="x < 0" />, the value of <InlineMath math="0^x" /> is undefined

This function is no longer an exponential function but rather constant at <InlineMath math="f(x) = 0" /> for <InlineMath math="x > 0" />. Then, because <InlineMath math="0^0" /> and <InlineMath math="0^x" /> for <InlineMath math="x < 0" /> are undefined, this function does not meet the definition of an exponential function.

<FunctionChart
  a={0}
  p={1}
  title="Constant Function"
  description={
    <>
      Line is always horizontal constant at <InlineMath math="y = 0" />, but
      undefined for <InlineMath math="x = 0" />.
    </>
  }
/>

## Examples of Exponential Functions

Here are some examples of exponential functions:

1. <InlineMath math="f(x) = 4^x" />

   This function has base number <InlineMath math="a = 4" /> and <InlineMath math="n = 1" />. Since <InlineMath math="a > 1" />, this function is monotonically increasing. The function value will get larger as x increases. For example, <InlineMath math="f(0) = 4^0 = 1" />, <InlineMath math="f(1) = 4^1 = 4" />, <InlineMath math="f(2) = 4^2 = 16" />.

2. <InlineMath math="f(x) = 3^{x+1}" />

   This function can be rewritten as <InlineMath math="f(x) = 3 \times 3^x" /> with base number <InlineMath math="a = 3" /> and <InlineMath math="n = 3" />. The graph of this function is also monotonically increasing, and the function value will get larger as x increases. For example, <InlineMath math="f(0) = 3^{0+1} = 3^1 = 3" />, <InlineMath math="f(1) = 3^{1+1} = 3^2 = 9" />.

3. <InlineMath math="f(x) = 5^{2x-1}" />

   This function has base number <InlineMath math="a = 5" /> with exponent <InlineMath math="2x-1" />. The function value will change more rapidly because the coefficient of x is 2. For example, <InlineMath math="f(0) = 5^{2(0)-1} = 5^{-1} = \frac{1}{5}" />, <InlineMath math="f(1) = 5^{2(1)-1} = 5^1 = 5" />.

4. <InlineMath math="f(x) = 0.5^x" />

   This function has base number <InlineMath math="a = 0.5" /> where <InlineMath math="0 < a < 1" />. This function is monotonically decreasing. The function value will get smaller as x increases. For example, <InlineMath math="f(0) = 0.5^0 = 1" />, <InlineMath math="f(1) = 0.5^1 = 0.5" />, <InlineMath math="f(2) = 0.5^2 = 0.25" />.

## Applications of Exponential Functions

Exponential functions are widely used in everyday life and various fields:

1. **Population Growth**: The number of bacteria reproducing can be modeled with an exponential function <InlineMath math="P(t) = P_0 \times 2^{t/n}" /> where <InlineMath math="P_0" /> is the initial number, t is time, and n is the time required for the population to double.

2. **Compound Interest**: If someone saves money with compound interest, the amount of savings after t years can be calculated with <InlineMath math="A(t) = P \times (1 + r)^t" /> where P is the initial principal and r is the interest rate.

3. **Radioactive Decay**: The amount of radioactive substance remaining after t years can be calculated with <InlineMath math="A(t) = A_0 \times 0.5^{t/h}" /> where <InlineMath math="A_0" /> is the initial amount and h is the half-life.

4. **Virus Spread**: The spread of disease in a population often follows an exponential model in the early phase.
