https://r2.erweima.ai/mathbot/image/1725842798447vxp12izg.jpg

The image contains the following mathematical problem:

**Function:**

\[
y = \sqrt{2x - 8}
\]

**Instructions:**

a) Determine and explain what type of function this is.

b) Domain.

c) Range.

d) Determine whether the function is increasing or decreasing, and over which intervals.

e) Determine whether the function is continuous or discontinuous.

f) Intercepts and asymptotes.

g) Create a table of values (at least 8 points), find the vertex, and graph the function.

---

### Let’s go step by step:

#### a) **Type of function:**
The given function is a **square root function**, as the variable \(x\) is inside a square root. This type of function generally produces a graph with a curved shape starting from a certain point and increasing or decreasing as \(x\) increases.

#### b) **Domain:**
For the square root function to be defined, the expression inside the square root must be non-negative. That is:

\[
2x - 8 \geq 0
\]

Solving for \(x\):

\[
x \geq 4
\]

Thus, the domain is:

\[
x \in [4, \infty)
\]

#### c) **Range:**
Since the square root function always outputs non-negative values, the range of the function is:

\[
y \in [0, \infty)
\]

#### d) **Increasing or decreasing intervals:**
The derivative of the function will help determine where the function is increasing or decreasing. The function is:

\[
y = \sqrt{2x - 8}
\]

The derivative \( \frac{dy}{dx} \) is:

\[
\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\sqrt{2x - 8}} \cdot 2 = \frac{1}{\sqrt{2x - 8}}
\]

Since \( \frac{1}{\sqrt{2x - 8}} > 0 \) for \( x \geq 4 \), the function is **increasing** on the interval \( [4, \infty) \).

#### e) **Continuity:**
The function is continuous on its domain \( x \geq 4 \). There are no breaks or discontinuities in the graph of the square root function.

#### f) **Intercepts and asymptotes:**
- **x-intercept:** Set \( y = 0 \):

\[
0 = \sqrt{2x - 8} \Rightarrow 2x - 8 = 0 \Rightarrow x = 4
\]

So, the x-intercept is at \( (4, 0) \).

- **y-intercept:** This function has no y-intercept because the domain does not include \( x = 0 \).

- **Asymptotes:** There are no vertical or horizontal asymptotes in this function.

#### g) **Table of values:**

To create a table, choose values of \(x\) starting from 4 (since it's the minimum value in the domain):

\[
\begin{array}{c|c}
x & y = \sqrt{2x - 8} \\
\hline
4 & 0 \\
5 & \sqrt{2(5) - 8} = \sqrt{2} \approx 1.41 \\
6 & \sqrt{2(6) - 8} = 2 \\
7 & \sqrt{2(7) - 8} = \sqrt{6} \approx 2.45 \\
8 & \sqrt{2(8) - 8} = 2.83 \\
9 & \sqrt{2(9) - 8} = \sqrt{10} \approx 3.16 \\
10 & \sqrt{2(10) - 8} = 4 \\
11 & \sqrt{2(11) - 8} = \sqrt{14} \approx 3.74 \\
\end{array}
\]

- **Vertex:** The point where the function starts is at \( (4, 0) \), which acts as the "vertex" of this increasing function.

---

### Do you have any specific questions, or would you like further details?

### 5 Related Questions:
1. How do you graph a square root function by hand?
2. What are the general properties of square root functions?
3. How does the graph of a square root function change with different coefficients?
4. What is the relationship between the domain of the function and the expression under the square root?
5. Can you explain how to compute the derivative of the square root function in more detail?

### Tip: Always check the domain first when working with square root functions to ensure you know where the function is defined!

Para la siguiente función: y = √(2x - 8), determine y argumente con los procedimientos respectivos: a) Qué tipo de función es. b) Dominio. c) Rango. d) Si es creciente y decreciente y en qué intervalos. e) Si es continua o discontinua. f) Intercepciones y asíntotas. g) Realice una tabla de valores (por lo menos 8), encuentre el vértice y grafique la función.

Detailed analysis of the square root function y = √(2x - 8), including domain, range, increasing/decreasing intervals, continuity, intercepts, asymptotes, and a vertex graph. This content is ideal for high school students learning advanced algebra concepts.

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