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The given function \( g(x) \) is defined as a piecewise function:

\[
g(x) = 
\begin{cases} 
x - 3 & \text{si } x < 0 \\
x^2 + 3 & \text{si } 0 \leq x < 2 \\
\sqrt{x - 2} & \text{si } x \geq 2 
\end{cases}
\]

### Let's break this down and then graph it step by step:

1. **For \( x < 0 \):** The function is \( g(x) = x - 3 \).
   - This is a linear function with a slope of 1 and a y-intercept of -3.
   - It will be a straight line passing through points like (-1, -4) and (-2, -5).

2. **For \( 0 \leq x < 2 \):** The function is \( g(x) = x^2 + 3 \).
   - This is a quadratic function that opens upwards with its vertex at \( (0, 3) \).
   - It will cover values from \( x = 0 \) to \( x = 2 \) and include points like \( (0, 3) \) and \( (1, 4) \).

3. **For \( x \geq 2 \):** The function is \( g(x) = \sqrt{x - 2} \).
   - This is a square root function that starts at \( x = 2 \) with \( g(2) = 0 \) and increases as \( x \) increases.
   - Points like \( (2, 0) \), \( (3, 1) \), and \( (4, 2) \) are on this curve.

### **Graph of the Function:**

Now, I'll plot the graph for you and analyze the domain and range.

Explore how to graph piecewise functions with detailed step-by-step examples. This guide covers linear, quadratic, and square root functions. Suitable for high school students and anyone learning about piecewise functions.

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