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evaluate the piecewise function for the given values of x

The given piecewise function is:

\[
f(x) = \begin{cases} 
2x + 1 & \text{if } x < 1 \\
-2x + 3 & \text{if } x \geq 1 
\end{cases}
\]

Now, let's evaluate the function for the given values of \( x \):

1. \( f(-2) \):
   - Since \( -2 < 1 \), we use the first part of the function: \( 2x + 1 \).
   \[
   f(-2) = 2(-2) + 1 = -4 + 1 = -3
   \]
   So, \( f(-2) = -3 \).

2. \( f(6) \):
   - Since \( 6 \geq 1 \), we use the second part of the function: \( -2x + 3 \).
   \[
   f(6) = -2(6) + 3 = -12 + 3 = -9
   \]
   So, \( f(6) = -9 \).

3. \( f(1) \):
   - Since \( 1 \geq 1 \), we use the second part of the function: \( -2x + 3 \).
   \[
   f(1) = -2(1) + 3 = -2 + 3 = 1
   \]
   So, \( f(1) = 1 \).

### Final Answers:
- \( f(-2) = -3 \)
- \( f(6) = -9 \)
- \( f(1) = 1 \)

Would you like more details on any of these evaluations? 

### Related Questions:
1. What would \( f(x) \) be for \( x = 0 \)?
2. How would the function change if the inequality for \( x \geq 1 \) was modified to \( x > 1 \)?
3. Can the function \( f(x) \) be continuous at \( x = 1 \)? Why or why not?
4. What is the domain and range of \( f(x) \)?
5. How would you graph this piecewise function?

### Tip:
For piecewise functions, always pay close attention to the inequalities to determine which part of the function to use for a specific value of \( x \).

Evaluate the piecewise function for the given values of x: f(-2), f(6), and f(1).

Learn how to evaluate a piecewise function for different values of x. The function is defined as f(x) = 2x + 1 for x < 1 and f(x) = -2x + 3 for x ≥ 1. This tutorial walks through the process of solving for f(-2), f(6), and f(1). Suitable for students in Grades 8-10.

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