Let's analyze the table step by step to find which function is best modeled by a **piecewise-linear function with two linear segments with different slopes**. ### Step 1: Understanding the Problem A piecewise-linear function with two segments means: - For the first part of the domain, the function has one constant rate of change (slope). - For the second part, it has a different constant rate of change (slope). - The change between the two slopes should be clear and abrupt, not gradual. ### Step 2: Examine Each Function #### **Change in \( f(x) \):** 2.1, 2.0, 2.1, 2.2, 2.1, 2.2, 2.0 - The rate of change is nearly constant throughout. - **Not piecewise-linear with two segments.** #### **Change in \( g(x) \):** 2.1, 3.2, 4.2, 5.1, 6.3, 7.2, 8.3 - The rate of change is increasing steadily. - **Not two constant segments.** #### **Change in \( h(x) \):** 2.1, 2.0, 2.1, 4.2, 4.1, 4.2, 4.0 - First three intervals: approximately 2.0 - Last four intervals: approximately 4.1 - **This shows two distinct constant slopes.** #### **Change in \( k(x) \):** 2.1, 3.2, 4.2, 5.1, 4.2, 3.2, 2.1 - Increases, then decreases symmetrically. - **Not two constant segments.** ### Step 3: Conclusion The function best modeled by a piecewise-linear function with two linear segments with different slopes is **h(x)**. ### **Final Answer** **h** --- **Explanation:** - For \( h(x) \), the first part (intervals 1–3) is about 2.0, and the second part (intervals 4–7) is about 4.1. - This matches the requirement for a piecewise-linear function with two segments with different slopes. **So, the answer is:** > **h**

Let's carefully analyze the problem step-by-step, considering each function's data, to determine which one can be modeled by a **piecewise-linear function with exactly two linear segments with different slopes**. --- ### **Given Data:** | **Interval of z** | **Change in f(z)** | **Change in g(z)** | **Change in h(z)** | **Change in k(z)** | |---------------------|--------------------|--------------------|--------------------|--------------------| | \( 0 < z < l \) | 2.1 | 2.1 | 2.1 | 2.1 | | \( l < z < 22 \) | 2.0 | 3.2 | 2.0 | 3.2 | | \( 22 < z < 33 \) | 2.1 | 4.2 | 2.1 | 4.2 | | \( 33 < z < dd \) | 22 | 5.1 | 42 | 5.1 | | \( dd < z < ib \) | 2.1 | 6.3 | 41 | 42 | | \( ib < z < bb \) | 2.2 | 7.2 | 12 | 3.2 | | \( bb < z < c \) | 2.0 | 8.3 | 1.0 | 2.1 | --- ### **Step 1: Understand the goal** We need to identify which function among \(f(z)\), \(g(z)\), \(h(z)\), \(k(z)\) **can be modeled by a piecewise-linear function with exactly two segments**, i.e., two different constant slopes, with a clear change point. - **Key idea:** - The function should have **two distinct regions** with **constant slopes** (i.e., constant rate of change), separated by a point where the slope changes abruptly. - The change in the rate should **not** be gradual; it should switch from one value to another. --- ### **Step 2: Analyze each function** --- ### **Function \(f(z)\):** | Change in \(f(z)\) | Values: 2.1, 2.0, 2.1, 22, 2.1, 2.2, 2.0 | - **Observation:** - First three changes are close to 2.0-2.1 (nearly constant). - At the fourth interval, there's a large jump: **22**. - After that, changes revert back to about 2.0-2.2. - **Conclusion:** - The presence of a **single large jump** at an interval suggests a **break** in the pattern. - **However**, because the initial segments are roughly similar (~2.0), then a sudden large change, then back to similar values, this **can** be modeled as a **piecewise-linear** with **two segments**: - First segment: slope ~2.0 - Second segment: slope possibly different after the jump. - **But**, the jump from ~2 to 22 is very abrupt, which could be modeled as a **discontinuity** or a change in slope. But since the large jump occurs only once, and the other parts are similar, it **may** fit the two-segment model. --- ### **Function \(g(z)\):** | Change in \(g(z)\) | Values: 2.1, 3.2, 4.2, 5.1, 6.3, 7.2, 8.3 | - The changes are gradually increasing: 2.1, 3.2, 4.2, 5.1, 6.3, 7.2, 8.3. - **Observation:** - The rate of change **increases steadily** over the intervals. - This suggests a **smooth increasing trend** in the rate of change, not just two constant segments. - **Conclusion:** - Since the rate of change is **not constant** in any segment but gradually increasing, it **cannot** be approximated as a **piecewise-linear with only two segments** with different slopes. - It resembles more of a **smooth curve**. --- ### **Function \(h(z)\):** | Change in \(h(z)\) | Values: 2.1, 2.0, 2.1, 42, 41, 12, 1.0 | - **Observation:** - First three values (~2.0) are similar, indicating a **constant slope** in the initial segment. - Then, there's a **large jump** at the fourth interval: 42, then 41, then decreasing to 12 and finally 1.0. - The pattern shows **two distinct regimes**: - **First regime:** slope ~2.0 (small changes) - **Second regime:** slope ~42 initially, then decreasing further. - **Conclusion:** - The initial segment is nearly constant slope (~2). - The second segment has a **much larger slope** (~42), then decreases. - The two segments have **distinct slopes**, separated at the point where the change jumps from around 2 to 42. - **Therefore**, \(h(z)\) is consistent with a **piecewise-linear function with two segments** of different slopes. --- ### **Function \(k(z)\):** | Change in \(k(z)\) | Values: 2.1, 3.2, 4.2, 5.1, 4.2, 3.2, 2.1 | - **Observation:** - The change starts increasing: 2.1, 3.2, 4.2, 5.1, then decreases symmetrically: 4.2, 3.2, 2.1. - This pattern is **not** two linear segments but rather a **symmetrical pattern**, like a **parabola** or **quadratic**. - **Conclusion:** - The pattern shows **more than two segments**, with the change rate increasing then decreasing symmetrically. - Not suitable for a **two-segment** piecewise-linear model. --- ### **Summary:** | Function | Suitability for **two-segment piecewise-linear model** | |------------|---------------------------------------------------------| | \(f(z)\) | **Partially suitable** due to one large jump, but not perfectly clear. | | \(g(z)\) | **Not suitable** (smoothly increasing slope). | | \(h(z)\) | **Best candidate**: clear two regimes with different slopes. | | \(k(z)\) | Not suitable (symmetrical pattern, more than two segments). | --- ### **Final conclusion:** **The function \(h(z)\) is the best modeled by a piecewise-linear function with two segments with different slopes.** --- ### **Answer:** **\(\boxed{h}\)** --- ### **Summary:** - \(f(z)\): Shows a possible jump but mainly similar slopes; not ideal. - \(g(z)\): Increasing rate of change; smooth trend, not two segments. - \(h(z)\): Two distinct regimes with different slopes; ideal candidate. - \(k(z)\): Symmetrical pattern, not just two segments. **Thus, the correct choice is: \(\boxed{\text{h}}\).**