# Nakafa Framework: LLM

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

Output docs content for large language models.

---

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

export const metadata = {
  title: "Square Root Function",
  description: "Explore square root functions with interactive graphs, domain/range analysis, transformations, and equations. Master graphing techniques through rocket height problems.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/18/2025",
  subject: "Functions and Their Modeling",
};

## Definition of Square Root Function

A square root function is a type of function that involves square root operations. This function has the general form <InlineMath math="f(x) = \sqrt{g(x)}" /> where <InlineMath math="g(x)" /> is the function inside the square root sign.

The simplest form of a square root function is <InlineMath math="f(x) = \sqrt{x}" />. This function takes an input value <InlineMath math="x" /> and produces the square root of that value.

### Characteristics of Square Root Functions

Square root functions have several special characteristics that distinguish them from other functions:

1. **Limited domain**: Since the square root of negative numbers is not defined in real numbers, the domain of square root functions is limited to values that make the expression inside the square root non-negative.

2. **Curved graph**: The graph of a square root function is a curve that starts from a certain point and continues to rise at an increasingly slower rate.

3. **Always non-negative values**: The result of a square root function is always non-negative (≥ 0).

## Domain and Range of Square Root Functions

To understand square root functions well, it's important to determine their domain and range.

### Determining Domain

The domain of a square root function <InlineMath math="f(x) = \sqrt{g(x)}" /> is all values of <InlineMath math="x" /> that make <InlineMath math="g(x) \geq 0" />.

**Steps to determine domain:**

| Step | Explanation                                             | Example: <InlineMath math="f(x) = \sqrt{x-2}" />                  |
| ---- | ------------------------------------------------------- | ----------------------------------------------------------------- |
| 1    | Identify the expression inside the square root          | <InlineMath math="g(x) = x-2" />                                  |
| 2    | Create the inequality <InlineMath math="g(x) \geq 0" /> | <InlineMath math="x-2 \geq 0" />                                  |
| 3    | Solve the inequality                                    | <InlineMath math="x \geq 2" />                                    |
| 4    | Write the domain                                        | <InlineMath math="D_f = \{x \mid x \geq 2, x \in \mathbb{R}\}" /> |

### Determining Range

The range of a square root function is all possible output values that the function can produce.

For the function <InlineMath math="f(x) = \sqrt{x-2}" />, since square roots always produce non-negative values, then:

<BlockMath math="R_f = \{y \mid y \geq 0, y \in \mathbb{R}\}" />

## Graph of Basic Square Root Function

Let's visualize the basic square root function <InlineMath math="f(x) = \sqrt{x}" />.

<LineEquation
  title={
    <>
      Basic Square Root Function: <InlineMath math="f(x) = \sqrt{x}" />
    </>
  }
  description="The graph of the basic square root function starts from point (0,0) and rises at an increasingly slower rate."
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i * 0.5;
        return { x, y: Math.sqrt(x), z: 0 };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
      labels: [{ text: "f(x) = √x", at: 5, offset: [0, 1, 0] }],
    },
  ]}
  showZAxis={false}
  cameraPosition={[10, 6, 10]}
/>

## Transformations of Square Root Functions

Square root functions can undergo various transformations that change the shape and position of their graphs.

### Horizontal Translation

The function <InlineMath math="f(x) = \sqrt{x-h}" /> shifts the graph of the basic square root function by <InlineMath math="h" /> units to the right (if <InlineMath math="h > 0" />) or to the left (if <InlineMath math="h < 0" />).

<LineEquation
  title="Horizontal Translation of Square Root Functions"
  description={
    <>
      Comparison of <InlineMath math="f(x) = \sqrt{x}" />,{" "}
      <InlineMath math="g(x) = \sqrt{x-2}" />, and{" "}
      <InlineMath math="h(x) = \sqrt{x+2}" />.
    </>
  }
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i * 0.5;
        return { x, y: Math.sqrt(x), z: 0 };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
      labels: [{ text: "√x", at: 5, offset: [0, -0.5, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = 2 + i * 0.5;
        return { x, y: Math.sqrt(x - 2), z: 0 };
      }),
      color: getColor("ORANGE"),
      showPoints: false,
      labels: [{ text: "√(x-2)", at: 10, offset: [0, -0.5, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = -2 + i * 0.5;
        return { x, y: Math.sqrt(x + 2), z: 0 };
      }),
      color: getColor("CYAN"),
      showPoints: false,
      labels: [{ text: "√(x+2)", at: 10, offset: [0, 0.5, 0] }],
    },
  ]}
  showZAxis={false}
  cameraPosition={[10, 6, 10]}
/>

### Vertical Translation

The function <InlineMath math="f(x) = \sqrt{x} + k" /> shifts the graph of the basic square root function by <InlineMath math="k" /> units upward (if <InlineMath math="k > 0" />) or downward (if <InlineMath math="k < 0" />).

| Function                                  | Transformation         | Domain                         | Range                           |
| ----------------------------------------- | ---------------------- | ------------------------------ | ------------------------------- |
| <InlineMath math="f(x) = \sqrt{x}" />     | Basic function         | <InlineMath math="x \geq 0" /> | <InlineMath math="y \geq 0" />  |
| <InlineMath math="f(x) = \sqrt{x} + 2" /> | Shift 2 units upward   | <InlineMath math="x \geq 0" /> | <InlineMath math="y \geq 2" />  |
| <InlineMath math="f(x) = \sqrt{x} - 3" /> | Shift 3 units downward | <InlineMath math="x \geq 0" /> | <InlineMath math="y \geq -3" /> |

### Dilation

The function <InlineMath math="f(x) = a\sqrt{x}" /> with <InlineMath math="a > 0" /> causes vertical dilation on the graph of the square root function.

<LineEquation
  title="Vertical Dilation of Square Root Functions"
  description={
    <>
      Comparison of <InlineMath math="f(x) = \sqrt{x}" />,{" "}
      <InlineMath math="g(x) = 2\sqrt{x}" />, and{" "}
      <InlineMath math="h(x) = \frac{1}{2}\sqrt{x}" />.
    </>
  }
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i * 0.5;
        return { x, y: Math.sqrt(x), z: 0 };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
      labels: [{ text: "√x", at: 5, offset: [0, -0.5, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i * 0.5;
        return { x, y: 2 * Math.sqrt(x), z: 0 };
      }),
      color: getColor("EMERALD"),
      showPoints: false,
      labels: [{ text: "2√x", at: 10, offset: [0, -0.5, 0] }],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = i * 0.5;
        return { x, y: 0.5 * Math.sqrt(x), z: 0 };
      }),
      color: getColor("AMBER"),
      showPoints: false,
      labels: [{ text: "½√x", at: 10, offset: [0, 0.5, 0] }],
    },
  ]}
  showZAxis={false}
  cameraPosition={[10, 6, 10]}
/>

## General Form of Square Root Functions

The general form of a square root function that undergoes transformations is:

<BlockMath math="f(x) = a\sqrt{b(x-h)} + k" />

Where:

- <InlineMath math="a" /> determines vertical dilation and reflection (if <InlineMath math="a < 0" />
  )
- <InlineMath math="b" /> determines horizontal dilation
- <InlineMath math="h" /> determines horizontal translation
- <InlineMath math="k" /> determines vertical translation

**Steps to draw the graph:**

| Step | Action                    | Example: <InlineMath math="f(x) = 2\sqrt{x-1} + 3" />                                          |
| ---- | ------------------------- | ---------------------------------------------------------------------------------------------- |
| 1    | Determine starting point  | <InlineMath math="x-1 = 0 \Rightarrow x = 1" />, <InlineMath math="f(1) = 3" />, Point: (1, 3) |
| 2    | Determine domain          | <InlineMath math="x-1 \geq 0 \Rightarrow x \geq 1" />                                          |
| 3    | Create value table        | Choose several values <InlineMath math="x \geq 1" />                                           |
| 4    | Calculate function values | For <InlineMath math="x = 2" />: <InlineMath math="f(2) = 2\sqrt{1} + 3 = 5" />                |
| 5    | Plot points               | Plot (1,3), (2,5), (5,7), etc.                                                                 |
| 6    | Connect points            | Create a smooth curve through the points                                                       |

## Square Root Function Equations

To solve equations involving square root functions, follow these steps:

| Step | Explanation             | Example: <InlineMath math="\sqrt{2x+3} = 5" />         |
| ---- | ----------------------- | ------------------------------------------------------ |
| 1    | Isolate the square root | Already isolated                                       |
| 2    | Square both sides       | <InlineMath math="(\sqrt{2x+3})^2 = 5^2" />            |
| 3    | Simplify                | <InlineMath math="2x+3 = 25" />                        |
| 4    | Solve                   | <InlineMath math="2x = 22 \Rightarrow x = 11" />       |
| 5    | Verify                  | <InlineMath math="\sqrt{2(11)+3} = \sqrt{25} = 5" /> ✓ |

## Square Root Function Inequalities

To solve square root function inequalities, pay attention to the domain and properties of square root functions.

**Example:** Solve <InlineMath math="\sqrt{x-1} < 3" />

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Condition: } x-1 \geq 0 \Rightarrow x \geq 1" />
  <BlockMath math="\sqrt{x-1} < 3" />
  <BlockMath math="x-1 < 9" />
  <BlockMath math="x < 10" />
</div>

Combining with the domain condition: <InlineMath math="1 \leq x < 10" />

## Practice Problems

1. Determine the domain and range of the function <InlineMath math="f(x) = \sqrt{3x-6}" />

2. Draw the graph of the function <InlineMath math="g(x) = -\sqrt{x+4} + 2" />

3. Solve the equation <InlineMath math="\sqrt{x+5} + \sqrt{x} = 5" />

4. A rocket is launched vertically. Its height after <InlineMath math="t" /> seconds is given by <InlineMath math="h(t) = 100\sqrt{t}" /> meters. What is the height of the rocket after 9 seconds?

5. Determine the value of <InlineMath math="x" /> that satisfies <InlineMath math="2\sqrt{x-3} \geq 6" />

### Answer Key

1. Domain: <InlineMath math="D_f = \{x \mid x \geq 2, x \in \mathbb{R}\}" />

   Range: <InlineMath math="R_f = \{y \mid y \geq 0, y \in \mathbb{R}\}" />

2. **Drawing the graph** <InlineMath math="g(x) = -\sqrt{x+4} + 2" />

   **Steps to draw:**

   | Step | Explanation              | Details for <InlineMath math="g(x) = -\sqrt{x+4} + 2" />                                                                                                   |
   | ---- | ------------------------ | ---------------------------------------------------------------------------------------------------------------------------------------------------------- |
   | 1    | Identify transformations | <InlineMath math="a = -1" /> (reflection across x-axis), <InlineMath math="h = -4" /> (shift 4 units left), <InlineMath math="k = 2" /> (shift 2 units up) |
   | 2    | Determine starting point | <InlineMath math="x+4 = 0 \Rightarrow x = -4" />, <InlineMath math="g(-4) = 0 + 2 = 2" />, Starting point: (-4, 2)                                         |
   | 3    | Determine domain         | <InlineMath math="x+4 \geq 0 \Rightarrow x \geq -4" />                                                                                                     |
   | 4    | Determine range          | Since <InlineMath math="a = -1 < 0" />, the graph decreases from the starting point, so <InlineMath math="y \leq 2" />                                     |
   | 5    | Create value table       | Choose values <InlineMath math="x \geq -4" />                                                                                                              |

   **Value table:**

   | <InlineMath math="x" /> | <InlineMath math="x+4" /> | <InlineMath math="\sqrt{x+4}" /> | <InlineMath math="-\sqrt{x+4}" /> | <InlineMath math="g(x) = -\sqrt{x+4} + 2" /> |
   | ----------------------- | ------------------------- | -------------------------------- | --------------------------------- | -------------------------------------------- |
   | -4                      | 0                         | 0                                | 0                                 | 2                                            |
   | -3                      | 1                         | 1                                | -1                                | 1                                            |
   | 0                       | 4                         | 2                                | -2                                | 0                                            |
   | 5                       | 9                         | 3                                | -3                                | -1                                           |
   | 12                      | 16                        | 4                                | -4                                | -2                                           |

   <LineEquation
     title={
       <>
         Graph of <InlineMath math="g(x) = -\sqrt{x+4} + 2" />
       </>
     }
     description="Graph of a square root function that undergoes reflection across the x-axis and translation."
     data={[
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = -4 + i * 0.8;
           const y = -Math.sqrt(x + 4) + 2;
           return { x, y, z: 0 };
         }),
         color: getColor("VIOLET"),
         showPoints: false,
         labels: [
           { text: "g(x) = -√(x+4) + 2", at: 8, offset: [0, -1, 0] },
           { text: "(-4, 2)", at: 0, offset: [0.5, 0.5, 0] },
         ],
       },
     ]}
     showZAxis={false}
     cameraPosition={[10, 6, 10]}
   />

3. <InlineMath math="x = 4" /> (Let <InlineMath math="u = \sqrt{x}" /> and <InlineMath math="v = \sqrt{x+5}" />
   )

4. <InlineMath math="h(9) = 100\sqrt{9} = 100 \times 3 = 300" /> meters

5. <InlineMath math="x \geq 12" /> (from <InlineMath math="\sqrt{x-3} \geq 3" /> and
   condition <InlineMath math="x \geq 3" />)
