The problem asks which function is best modeled by a piecewise-linear function with two linear segments with different slopes based on the average rates of change shown in the table. We need to look for a function whose rate of change remains relatively constant in one section of the intervals and then changes sharply in another section.

Analyzing Each Function:

1. $f(x)$:

   • The rates are fairly consistent: $2.1, 2.0, 2.1, 2.2, 2.1, 2.2, 2.0$.
   • This indicates a near-constant slope, so $f(x)$ is likely a straight line (not piecewise).

2. $g(x)$:

   • The rates increase progressively: $2.1, 3.2, 4.2, 5.1, 6.3, 7.2, 8.3$.
   • This suggests that $g(x)$ has a single increasing slope, not two distinct segments.

3. $h(x)$:

   • The rates are steady for the first three intervals: $2.1, 2.0, 2.1$, and then jump: $4.2, 4.1, 4.2, 4.0$.
   • This shows two distinct segments: one with a slope around 2.1 and the other around 4.1. This is a good candidate for piecewise linearity.

4. $k(x)$:

   • The rates change more steadily: $2.1, 3.2, 4.2, 5.1, 4.2, 3.2, 2.1$.
   • There seems to be no sharp change indicating two linear segments; rather, it suggests a peak and a decrease.

Conclusion:

• $h(x)$ is the function that is best modeled by a piecewise-linear function with two linear segments with different slopes: one segment with a slope around 2.1 and another around 4.1.

Would you like a more detailed breakdown or further questions? Here are some related questions:

1. What is the definition of a piecewise-linear function?
2. How can we interpret average rates of change in terms of slope?
3. Why is a function with nearly constant rates of change not a good candidate for piecewise-linear modeling?
4. How do changing rates of change indicate different linear segments?
5. Can you describe how to model a real-world scenario with a piecewise-linear function?

Tip: In piecewise-linear functions, look for intervals with distinct changes in slope, as this indicates the "pieces" or segments where the function behaves differently.