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The function \( f(x) \) is given piecewise as:

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

We need to calculate \( f(-2) \), \( f(1) \), and \( f(2) \).

### 1. \( f(-2) \):
Since \( -2 < 1 \), we use the first piece of the function \( f(x) = x^2 + 2 \).

\[
f(-2) = (-2)^2 + 2 = 4 + 2 = 6
\]

### 2. \( f(1) \):
Since \( 1 \geq 1 \), we use the second piece of the function \( f(x) = 2x^2 + 2 \).

\[
f(1) = 2(1)^2 + 2 = 2(1) + 2 = 2 + 2 = 4
\]

### 3. \( f(2) \):
Since \( 2 \geq 1 \), we use the second piece of the function \( f(x) = 2x^2 + 2 \).

\[
f(2) = 2(2)^2 + 2 = 2(4) + 2 = 8 + 2 = 10
\]

### Final answers:
- \( f(-2) = 6 \)
- \( f(1) = 4 \)
- \( f(2) = 10 \)

Would you like more details or have any questions?

Here are 5 relative questions to expand:
1. What is the significance of piecewise functions in real-world modeling?
2. How does continuity at \( x = 1 \) apply to this function?
3. Can a piecewise function be both continuous and differentiable everywhere?
4. What are the criteria to determine which part of a piecewise function to use?
5. How can we graph this function and its behavior near \( x = 1 \)?

**Tip:** Always check the conditions on \( x \) carefully when working with piecewise functions to ensure you're applying the correct expression.

Given the function f(x) as a piecewise function, calculate f(-2), f(1), and f(2).

Learn how to calculate values of a piecewise function, specifically f(-2), f(1), and f(2). This involves applying quadratic functions depending on the value of x, suitable for students in Grades 8-10.

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