To understand the function \( f(x) \), we need to break it down according to its piecewise definition:

\[ f(x) = 
\begin{cases} 
x - 2 & \text{if } x \leq 0 \\
2x + 1 & \text{if } x > 0 
\end{cases} \]

Let's analyze each part separately.

### For \( x \leq 0 \):
When \( x \) is less than or equal to zero, the function is defined as:
\[ f(x) = x - 2 \]

### For \( x > 0 \):
When \( x \) is greater than zero, the function is defined as:
\[ f(x) = 2x + 1 \]

### Examples:
1. If \( x = -1 \):
   Since \( -1 \leq 0 \), we use the first part of the function:
   \[ f(-1) = -1 - 2 = -3 \]

2. If \( x = 2 \):
   Since \( 2 > 0 \), we use the second part of the function:
   \[ f(2) = 2(2) + 1 = 4 + 1 = 5 \]

3. If \( x = 0 \):
   Since \( 0 \leq 0 \), we use the first part of the function:
   \[ f(0) = 0 - 2 = -2 \]

Would you like a graphical representation of this function or more detailed information about how it behaves for different values of \( x \)?

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### Questions for Further Exploration:
1. How does the function \( f(x) \) change when \( x \) transitions from negative to positive?
2. What is the value of \( f(x) \) at \( x = 0 \)?
3. How would you describe the behavior of \( f(x) \) as \( x \) approaches 0 from the left?
4. How would you describe the behavior of \( f(x) \) as \( x \) approaches 0 from the right?
5. What are the intercepts of \( f(x) \) with the x-axis and y-axis?
6. How would you plot \( f(x) \) on a coordinate plane?
7. Can you find the domain and range of \( f(x) \)?
8. How would the function change if the conditions for \( x \) were altered?

### Tip:
When dealing with piecewise functions, carefully analyze each segment of the function separately and consider how they connect at the boundaries.

Explore the piecewise function f(x) = {x - 2, if x ≤ 0; 2x + 1, if x > 0}. Analyze its behavior for different values of x, including examples and graphical representation. Suitable for high school students.

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