To interpret the graph's key function for the domain interval \(9 < t < 14\), we first need to understand the context provided by the choices given. Since the options suggest scenarios involving running, cycling, resting, and driving, we want to determine which of these scenarios would fit with a portion of a piecewise function that represents a linear segment, implying consistent or steady action over time.
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A runner sprinting at a constant rate starting at minute nine: This option suggests that the runner is actively running during this period, which would correlate to a linear increase in distance over time.
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A cyclist taking a cool down ride at a slower constant pace: Similar to the runner, this option indicates a consistent activity over the time interval, possibly at a lower rate.
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A runner resting for 5 minutes: If the function were to represent rest, it may display a flat line (indicating no movement), which would not correlate with a constant rate of speed.
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Driving a car at a constant rate for five hours: This option implies a constant speed but does not fit well within the specified time interval from 9 to 14 (which seems more likely to represent minutes rather than hours).
Given this analysis, the most logical scenarios are either the runner or cyclist who is moving at a constant pace. However, since the first two options specify "starting at minute nine," both could potentially apply, but they describe different activities.
If we were to strictly interpret "the key function of the graph," it would likely correspond to either the runner or cyclist moving steadily. Still, since the context suggests movement, and considering how intensity may differ, a specific runner sprinting corresponds more closely with a constant rate at a given intensity.
Conclusion: A runner sprinting at a constant rate starting at minute nine is the most fitting scenario for the portion of the piecewise function over the interval \(9 < t < 14\), as it directly correlates to a linear increase in distance over time.