https://file.aiquickdraw.com/mathbot/file/1730050413903f6kyx2do.jpg

Let's analyze the table of values to determine the appropriate piecewise function.

### Table Analysis
The table provides the following values:
\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\
\hline
f(x) & -1 & 2 & 5 & 6 & 20 & 20 & 42 \\
\hline
\end{array}
\]

### Observing the Behavior
- For \( x = -2, -1, 0, 1 \): The function values change by relatively small amounts. This suggests a **linear** or **quadratic** pattern in this interval.
- For \( x = 2, 3, 4 \): The function values increase rapidly, which could indicate a **quadratic** or **cubic** behavior.

### Option Evaluation
Based on this, let's examine the options:
1. **Linear, then cubic**: This is possible since the early part could be linear, and the rapid increase could be modeled by a cubic function.
2. **Cubic, then linear**: This does not match because the function’s behavior starts with smaller increases (suggesting linear), not cubic.
3. **Linear, then quadratic**: This also seems reasonable if the initial values follow a linear pattern, and the rapid increase could be modeled by a quadratic function.
4. **Quadratic, then linear**: This does not match since the function does not flatten after increasing.

### Conclusion
The best matches are:
- **Option 1**: Linear, then cubic
- **Option 3**: Linear, then quadratic

Given the rapid growth from \( x = 2 \) onward, **Option 3 (Linear and then Quadratic)** seems most consistent with the data pattern in the table.

### Final Answer
The answer is likely:
**"A piecewise function consisting of a linear function and then a quadratic function."**

Would you like further explanation or have additional questions?

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#### Related Questions
1. How can we tell if a function is linear, quadratic, or cubic just from a table?
2. What are the characteristics of piecewise functions, and how are they applied in real-life situations?
3. Can a function have both quadratic and cubic components in one model?
4. How do the properties of linear, quadratic, and cubic functions differ?
5. What tools can be used to determine the best-fit model for data?

#### Tip
When analyzing a table for patterns, look for consistent differences (linear), changes in differences (quadratic), or higher-level variations (cubic or higher).

Which of the following functions models the data in the table?

The table gives selected data points from a data set:

| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|---|---|---|---|---|
| f(x) | -1 | 2 | 5 | 6 | 20 | 20 | 42 |

Options:
1. A piecewise function consisting of a linear function and then a cubic function
2. A piecewise function consisting of a cubic function and then a linear function
3. A piecewise function consisting of a linear function and then a quadratic function
4. A piecewise function consisting of a quadratic function and then a linear function

Explore how to model a data set using a piecewise function with linear and quadratic components. Given specific values in a table, analyze the best-fit functions for each section. This problem introduces core concepts in piecewise functions and polynomial modeling, ideal for high school students.

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