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URL: https://nakafa.com/en/subjects/mathematics/function-modeling/square-root-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/function-modeling/square-root-function/en.mdx

Study square root functions with graphs, domain and range, transformations, equations, and a rocket height problem.

---

## 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 $$f(x) = \sqrt{g(x)}$$ where $$g(x)$$ is the function inside the square root sign.

Visible text: A square root function is a type of function that involves square root operations. This function has the general form where is the function inside the square root sign.

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

Visible text: The simplest form of a square root function is . This function takes an input value 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 ($$\geq 0$$).

Visible text: 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 ().

## 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 $$f(x) = \sqrt{g(x)}$$ is all values of $$x$$ that make $$g(x) \geq 0$$.

Visible text: The domain of a square root function is all values of that make .

**Steps to determine domain:**

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

Visible text: | Step | Explanation | Example: |
| ---- | ------------------------------------------------------- | ----------------------------------------------------------------- |
| | Identify the expression inside the square root | |
| | Create the inequality | |
| | Solve the inequality | |
| | Write the domain | |

### Determining Range

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

For the function $$f(x) = \sqrt{x-2}$$, since square roots always produce non-negative values, then:

Visible text: For the function , since square roots always produce non-negative values, then:

```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 $$f(x) = \sqrt{x}$$.

Visible text: Let's visualize the basic square root function .

Component: LineEquation
Props:
- title: Basic Square Root Function: $$f(x) = \sqrt{x}$$
  Visible text: Basic Square Root Function:
- description: The graph of the basic square root function starts from point $$(0,0)$$ and rises at an increasingly slower rate.
  Visible text: The graph of the basic square root function starts from point 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 $$f(x) = \sqrt{x-h}$$ shifts the graph of the basic square root function by $$h \text{ units}$$ to the right (if $$h > 0$$) or to the left (if $$h < 0$$).

Visible text: The function shifts the graph of the basic square root function by to the right (if ) or to the left (if ).

Component: LineEquation
Props:
- title: Horizontal Translation of Square Root Functions
- description: Comparison of $$f(x) = \sqrt{x}$$,{" "}
$$g(x) = \sqrt{x-2}$$, and{" "}
$$h(x) = \sqrt{x+2}$$.
  Visible text: Comparison of ,{" "}
, and{" "}
.
- 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 $$f(x) = \sqrt{x} + k$$ shifts the graph of the basic square root function by $$k \text{ units}$$ upward (if $$k > 0$$) or downward (if $$k < 0$$).

Visible text: The function shifts the graph of the basic square root function by upward (if ) or downward (if ).

| Function                                  | Transformation         | Domain                         | Range                           |
| ----------------------------------------- | ---------------------- | ------------------------------ | ------------------------------- |
| $$f(x) = \sqrt{x}$$     | Basic function         | $$x \geq 0$$ | $$y \geq 0$$  |
| $$f(x) = \sqrt{x} + 2$$ | Shift $$2 \text{ units}$$ upward   | $$x \geq 0$$ | $$y \geq 2$$  |
| $$f(x) = \sqrt{x} - 3$$ | Shift $$3 \text{ units}$$ downward | $$x \geq 0$$ | $$y \geq -3$$ |

Visible text: | Function | Transformation | Domain | Range |
| ----------------------------------------- | ---------------------- | ------------------------------ | ------------------------------- |
| | Basic function | | |
| | Shift upward | | |
| | Shift downward | | |

### Dilation

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

Visible text: The function with causes vertical dilation on the graph of the square root function.

Component: LineEquation
Props:
- title: Vertical Dilation of Square Root Functions
- description: Comparison of $$f(x) = \sqrt{x}$$,{" "}
$$g(x) = 2\sqrt{x}$$, and{" "}
$$h(x) = \frac{1}{2}\sqrt{x}$$.
  Visible text: Comparison of ,{" "}
, and{" "}
.
- 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:

```math
f(x) = a\sqrt{b(x-h)} + k
```

Where:

- $$a$$ determines vertical dilation and reflection (if $$a < 0$$
  )
- $$b$$ determines horizontal dilation
- $$h$$ determines horizontal translation
- $$k$$ determines vertical translation

Visible text: - determines vertical dilation and reflection (if 
 )
- determines horizontal dilation
- determines horizontal translation
- determines vertical translation

**Steps to draw the graph:**

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

Visible text: | Step | Action | Example: |
| ---- | ------------------------- | ---------------------------------------------------------------------------------------------- |
| | Determine starting point | , , Point: |
| | Determine domain | |
| | Create value table | Choose several values |
| | Calculate function values | For : |
| | Plot points | Plot , , , etc. |
| | 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: $$\sqrt{2x+3} = 5$$         |
| ---- | ----------------------- | ------------------------------------------------------ |
| $$1$$ | Isolate the square root | Already isolated                                       |
| $$2$$ | Square both sides       | $$(\sqrt{2x+3})^2 = 5^2$$            |
| $$3$$ | Simplify                | $$2x+3 = 25$$                        |
| $$4$$ | Solve                   | $$2x = 22 \Rightarrow x = 11$$       |
| $$5$$ | Verify                  | $$\sqrt{2(11)+3} = \sqrt{25} = 5$$ ✓ |

Visible text: | Step | Explanation | Example: |
| ---- | ----------------------- | ------------------------------------------------------ |
| | Isolate the square root | Already isolated |
| | Square both sides | |
| | Simplify | |
| | Solve | |
| | Verify | ✓ |

## Square Root Function Inequalities

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

**Example:** Solve $$\sqrt{x-1} < 3$$

Visible text: **Example:** Solve

Component: MathContainer
Children:

```math
\text{Condition: } x-1 \geq 0 \Rightarrow x \geq 1
```

```math
\sqrt{x-1} < 3
```

```math
x-1 < 9
```

```math
x < 10
```

Combining with the domain condition: $$1 \leq x < 10$$

Visible text: Combining with the domain condition:

## Practice Problems

1. Determine the domain and range of the function $$f(x) = \sqrt{3x-6}$$

2. Draw the graph of the function $$g(x) = -\sqrt{x+4} + 2$$

3. Solve the equation $$\sqrt{x+5} + \sqrt{x} = 5$$

4. A rocket is launched vertically. Its height after $$t \text{ seconds}$$ is given by $$h(t) = 100\sqrt{t} \text{ meters}$$. What is the height of the rocket after $$9 \text{ seconds}$$?

5. Determine the value of $$x$$ that satisfies $$2\sqrt{x-3} \geq 6$$

Visible text: 1. Determine the domain and range of the function 

2. Draw the graph of the function 

3. Solve the equation 

4. A rocket is launched vertically. Its height after is given by . What is the height of the rocket after ?

5. Determine the value of that satisfies

### Answer Key

1. Domain: $$D_f = \{x \mid x \geq 2, x \in \mathbb{R}\}$$

   Range: $$R_f = \{y \mid y \geq 0, y \in \mathbb{R}\}$$

2. **Drawing the graph** $$g(x) = -\sqrt{x+4} + 2$$

   **Steps to draw:**

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

   **Value table:**

   | $$x$$ | $$x+4$$ | $$\sqrt{x+4}$$ | $$-\sqrt{x+4}$$ | $$g(x) = -\sqrt{x+4} + 2$$ |
   | ----------------------- | ------------------------- | -------------------------------- | --------------------------------- | -------------------------------------------- |
   | $$-4$$ | $$0$$ | $$0$$ | $$0$$ | $$2$$ |
   | $$-3$$ | $$1$$ | $$1$$ | $$-1$$ | $$1$$ |
   | $$0$$ | $$4$$ | $$2$$ | $$-2$$ | $$0$$ |
   | $$5$$ | $$9$$ | $$3$$ | $$-3$$ | $$-1$$ |
   | $$12$$ | $$16$$ | $$4$$ | $$-4$$ | $$-2$$ |

   <LineEquation
     title={
       <>
         Graph of $$g(x) = -\sqrt{x+4} + 2$$
       </>
     }
     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. $$x = 4$$ (Let $$u = \sqrt{x}$$ and $$v = \sqrt{x+5}$$
   )

4. $$h(9) = 100\sqrt{9} = 100 \times 3 = 300 \text{ meters}$$

5. $$x \geq 12$$ (from $$\sqrt{x-3} \geq 3$$ and
   condition $$x \geq 3$$)

Visible text: 1. Domain: 

 Range: 

2. **Drawing the graph** 

 **Steps to draw:**

 | Step | Explanation | Details for |
 | ---- | ------------------------ | ---------------------------------------------------------------------------------------------------------------------------------------------------------- |
| | Identify transformations | (reflection across the -axis), (shift left), (shift up) |
| | Determine starting point | , , Starting point: |
 | | Determine domain | |
 | | Determine range | Since , the graph decreases from the starting point, so |
 | | Create value table | Choose values |

 **Value table:**

 | | | | | |
 | ----------------------- | ------------------------- | -------------------------------- | --------------------------------- | -------------------------------------------- |
 | | | | | |
 | | | | | |
 | | | | | |
 | | | | | |
 | | | | | |

 <LineEquation
 title={
 <>
 Graph of 
 </>
 }
 description={<>Graph of a square root function that undergoes reflection across the -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. (Let and 
 )

4. 

5. (from and
 condition )