https://file.aiquickdraw.com/mathbot/file/1727714887712v53ak0q4.jpg

The table provides values for a function \( f(x) \) based on different values of \( x \). We are tasked with determining which type of piecewise function best models the data.

### Data points from the table:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 1 \\
-1 & 3 \\
0 & 5 \\
1 & 8 \\
2 & 18 \\
3 & 32 \\
4 & 50 \\
\hline
\end{array}
\]

### Observations:
- For \( x = -2, -1, 0, 1 \), the function appears to follow a linear pattern: \( f(x) = 2x + 5 \), with slope 2.
- For \( x = 2, 3, 4 \), the function seems to grow faster, suggesting a cubic behavior.

### Testing possible answers:
1. **A piecewise function consisting of a linear function and then a quadratic function**:
   - Unlikely because the growth from \( x = 2 \) to \( x = 4 \) suggests a higher-order polynomial, not just a quadratic.

2. **A piecewise function consisting of a linear function and then a cubic function**:
   - This fits, as the data points initially suggest linear growth, followed by cubic growth from \( x = 2 \) onwards.

3. **A piecewise function consisting of a cubic function and then a linear function**:
   - This does not fit because the function is initially linear, not cubic.

4. **A piecewise function consisting of a quadratic function and then a linear function**:
   - This does not fit the data for the same reason as answer (1), and the transition at \( x = 2 \) indicates cubic growth, not quadratic.

### Conclusion:
The correct answer is **"A piecewise function consisting of a linear function and then a cubic function."** This matches the selected option in the image.

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Would you like further details or have any questions?

### Related Questions:
1. How do you identify the linear and cubic parts of a piecewise function from a table of values?
2. What are the key characteristics of cubic functions in comparison to quadratic functions?
3. How can you verify whether a function is linear or higher order based on its growth rate?
4. How do you write the piecewise function explicitly for the given data?
5. What is the general approach to solving problems involving piecewise functions?

### Tip:
Always check the rate of change between consecutive values in a table to detect if the function switches from linear to a higher-degree polynomial.

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

Explore how to model data using a piecewise function consisting of linear and cubic functions. The problem involves analyzing data points to identify different patterns and selecting the correct mathematical model. Suitable for students in Grades 9-12.

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