# Nakafa Framework: LLM

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

Output docs content for large language models.

---

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

export const metadata = {
  title: "Multiplication and Division of Functions",
  description: "Master function multiplication and division: domain rules, zero restrictions, and practical examples. Solve (f·g)(x) and (f/g)(x) with confidence.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/27/2025",
  subject: "Function Composition and Inverse Function",
};

## Multiplication of Two Functions

Multiplying two functions, <InlineMath math="f" /> and <InlineMath math="g" />, is as easy as multiplying two numbers. We just multiply the result of <InlineMath math="f(x)" /> by <InlineMath math="g(x)" /> for the same value of <InlineMath math="x" />. The result is a new function <InlineMath math="(f \cdot g)" />.

<div className="flex flex-col gap-4">
  <LineEquation
    title="Function Multiplication Visualization"
    description={
      <>
        Observe how the lines <InlineMath math="f(x)=x" /> and{" "}
        <InlineMath math="g(x)=2" /> are multiplied to become{" "}
        <InlineMath math="(f \cdot g)(x)=2x" />.
      </>
    }
    data={[
      {
        points: Array.from({ length: 11 }, (_, i) => ({
          x: i - 5,
          y: i - 5,
          z: 0,
        })),
        color: getColor("ORANGE"),
        labels: [{ text: "f(x)=x", at: 6, offset: [1, -0.5, 0] }],
      },
      {
        points: Array.from({ length: 11 }, (_, i) => ({ x: i - 5, y: 2, z: 0 })),
        color: getColor("SKY"),
        labels: [{ text: "g(x)=2", at: 8, offset: [1, 0.5, 0] }],
      },
      {
        points: Array.from({ length: 11 }, (_, i) => ({
          x: i - 5,
          y: 2 * (i - 5),
          z: 0,
        })),
        color: getColor("ROSE"),
        labels: [{ text: "(f⋅g)(x)=2x", at: 7, offset: [-1.5, 1, 0] }],
      },
    ]}
  />

  <BlockMath math="(f \cdot g)(x) = f(x) \cdot g(x)" />
</div>

Just like addition and subtraction, this multiplication machine <InlineMath math="(f \cdot g)" /> can only process raw materials (values of <InlineMath math="x" />) that can be processed by _both_ original machines, <InlineMath math="f" /> and <InlineMath math="g" />. So, its domain is the intersection of the domain of <InlineMath math="f" /> and the domain of <InlineMath math="g" />.

<BlockMath math="D_{f \cdot g} = D_f \cap D_g" />

### Example of Multiplication

Let's use slightly different functions this time:

1.  <InlineMath math="f(x) = x^2" />, with domain <InlineMath math="D_f = \{x | x \in \mathbb{R}\}" /> (all
    real numbers).
2.  <InlineMath math="g(x) = x - 1" />, with domain <InlineMath math="D_g = \{x | x \in \mathbb{R}\}" /> (all
    real numbers).

**Step 1: Determine the resulting function from multiplication**

<div className="flex flex-col gap-4">
  <BlockMath math="(f \cdot g)(x) = f(x) \cdot g(x)" />
  <BlockMath math="= (x^2) \cdot (x - 1)" />
  <BlockMath math="= x^3 - x^2" />
</div>

**Step 2: Determine the domain of the resulting function**

We find the intersection of <InlineMath math="D_f" /> and <InlineMath math="D_g" />:

<div className="flex flex-col gap-4">
  <BlockMath math="D_{f \cdot g} = D_f \cap D_g" />
  <BlockMath math="= \{x | x \in \mathbb{R}\} \cap \{x | x \in \mathbb{R}\}" />
  <BlockMath math="= \{x | x \in \mathbb{R}\}" />
</div>

So, the resulting function from multiplication is <InlineMath math="(f \cdot g)(x) = x^3 - x^2" /> with the domain of all real numbers.

## Division of Two Functions

Dividing function <InlineMath math="f" /> by function <InlineMath math="g" /> is also similar: we divide the result of <InlineMath math="f(x)" /> by <InlineMath math="g(x)" />. The result is a new function <InlineMath math="(\frac{f}{g})" />.

<BlockMath math="\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}" />

Now, there's a **very important additional rule!** We know that division by zero is not allowed. So, besides the value of <InlineMath math="x" /> needing to be in the domain of both <InlineMath math="f" /> and <InlineMath math="g" />, the value of <InlineMath math="g(x)" /> (the divisor function) **cannot be equal to zero**.

Therefore, the domain of the division function <InlineMath math="(\frac{f}{g})" /> is the intersection of domains <InlineMath math="D_f" /> and <InlineMath math="D_g" />, but we must _exclude_ all values of <InlineMath math="x" /> that cause <InlineMath math="g(x) = 0" />.

<BlockMath math="D_{\frac{f}{g}} = D_f \cap D_g - \{x | g(x) = 0\}" />

The <InlineMath math="-" /> sign here means "minus" or "excluded".

### Example of Division

We use the same functions as in the multiplication example:

1.  <InlineMath math="f(x) = x^2" />, <InlineMath math="D_f = \{x | x \in \mathbb{R}\}" />
2.  <InlineMath math="g(x) = x - 1" />, <InlineMath math="D_g = \{x | x \in \mathbb{R}\}" />

**Step 1: Determine the resulting function from division**

<BlockMath math="\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2}{x-1}" />

**Step 2: Determine the domain of the resulting function**

First, find the intersection of <InlineMath math="D_f" /> and <InlineMath math="D_g" />:

<BlockMath math="D_f \cap D_g = \{x | x \in \mathbb{R}\}" />

Second, find the value of <InlineMath math="x" /> that makes <InlineMath math="g(x) = 0" />:

<div className="flex flex-col gap-4">
  <BlockMath math="g(x) = 0" />
  <BlockMath math="x - 1 = 0" />
  <BlockMath math="x = 1" />
</div>

Third, exclude the value <InlineMath math="x=1" /> from the intersection of the domains:

<BlockMath math="D_{\frac{f}{g}} = \{x | x \in \mathbb{R}\} - \{1\}" />

Or it can also be written as:

<BlockMath math="D_{\frac{f}{g}} = \{x | x \in \mathbb{R}, x \neq 1\}" />

So, the resulting function from division is <InlineMath math="(\frac{f}{g})(x) = \frac{x^2}{x-1}" /> with the domain of all real numbers except <InlineMath math="x=1" />.

## Practice Problems

Given the function <InlineMath math="f(x) = \sqrt{x+4}" /> with <InlineMath math="D_f = \{x | x \ge -4, x \in \mathbb{R}\}" /> and function <InlineMath math="g(x) = x^2 - 9" /> with <InlineMath math="D_g = \{x | x \in \mathbb{R}\}" />.

1.  Determine <InlineMath math="(f \cdot g)(x)" /> and its domain <InlineMath math="D_{f \cdot g}" />.
2.  Determine <InlineMath math="(\frac{f}{g})(x)" /> and its domain <InlineMath math="D_{\frac{f}{g}}" />.
3.  Calculate the value of <InlineMath math="(f \cdot g)(5)" />.
4.  Is <InlineMath math="(\frac{f}{g})(3)" /> defined? Explain.

### Answer Key

1.  **Finding <InlineMath math="(f \cdot g)(x)" />:**

    <div className="flex flex-col gap-4">
      <BlockMath math="(f \cdot g)(x) = f(x) \cdot g(x)" />
      <BlockMath math="= (\sqrt{x+4}) \cdot (x^2 - 9)" />
      <BlockMath math="= (x^2 - 9)\sqrt{x+4}" />
    </div>

    **Finding Domain <InlineMath math="D_{f \cdot g}" />:**

    <div className="flex flex-col gap-4">
      <BlockMath math="D_{f \cdot g} = D_f \cap D_g" />
      <BlockMath math="= \{x | x \ge -4, x \in \mathbb{R}\} \cap \{x | x \in \mathbb{R}\}" />
      <BlockMath math="= \{x | x \ge -4, x \in \mathbb{R}\}" />
    </div>

    So, <InlineMath math="(f \cdot g)(x) = (x^2 - 9)\sqrt{x+4}" /> with domain <InlineMath math="\{x | x \ge -4, x \in \mathbb{R}\}" />.

2.  **Finding <InlineMath math="(\frac{f}{g})(x)" />:**

    <BlockMath math="\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x+4}}{x^2 - 9}" />

    **Finding Domain <InlineMath math="D_{\frac{f}{g}}" />:**

    Domain intersection: <InlineMath math="D_f \cap D_g = \{x | x \ge -4, x \in \mathbb{R}\}" />.

    Find <InlineMath math="x" /> that makes <InlineMath math="g(x)=0" />:

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

    Exclude <InlineMath math="x=3" /> and <InlineMath math="x=-3" /> from the domain intersection:

    <BlockMath math="D_{\frac{f}{g}} = \{x | x \ge -4, x \in \mathbb{R}\} - \{-3, 3\}" />

    Or it can be written as:

    <BlockMath math="D_{\frac{f}{g}} = \{x | x \ge -4, x \in \mathbb{R}, x \neq -3, x \neq 3\}" />

    So, <InlineMath math="(\frac{f}{g})(x) = \frac{\sqrt{x+4}}{x^2 - 9}" /> with domain <InlineMath math="\{x | x \ge -4, x \neq -3, x \neq 3\}" />.

3.  **Calculating <InlineMath math="(f \cdot g)(5)" />:**

    We use the result from number 1: <InlineMath math="(f \cdot g)(x) = (x^2 - 9)\sqrt{x+4}" />.

    Since <InlineMath math="5 \ge -4" />, <InlineMath math="x=5" /> is in the domain <InlineMath math="D_{f \cdot g}" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="(f \cdot g)(5) = (5^2 - 9)\sqrt{5+4}" />
      <BlockMath math="= (25 - 9)\sqrt{9}" />
      <BlockMath math="= (16)(3)" />
      <BlockMath math="= 48" />
    </div>

4.  **Is <InlineMath math="(\frac{f}{g})(3)" /> defined?**

    Undefined. We look at the domain of <InlineMath math="(\frac{f}{g})(x)" /> from number 2, which is <InlineMath math="\{x | x \ge -4, x \neq -3, x \neq 3\}" />. The value <InlineMath math="x=3" /> is explicitly excluded from the domain because it would cause the denominator <InlineMath math="g(x) = x^2 - 9" /> to become zero (<InlineMath math="3^2 - 9 = 0" />). Division by zero is not allowed in mathematics.