Let's analyze the functions specified in the problem:
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Function f(t): This represents the height of the rocket at time \( t \). It will typically be a function that increases to a maximum height and then decreases as the rocket falls back to the ground, essentially describing a projectile's motion under the influence of gravity.
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Function g(t): This is described as a constant function representing the height of the platform. If \( g(t) = h \) where \( h \) is a constant (the height of the platform), then this value does not change with time.
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Function f(t) + g(t): The sum \( f(t) + g(t) \) essentially gives the total height of the rocket above the ground level (the height of the platform plus the height of the rocket relative to the platform). Therefore, this function describes how high the rocket is above the ground at any time \( t \).
Description of f(t) + g(t):
- f(t) represents the vertical position of the rocket relative to the platform.
- g(t), being constant, adds a fixed height to the rocket's height during its flight.
- The resulting function \( h(t) = f(t) + g(t) \) describes the absolute height of the rocket above ground level.
Domain of f(t) + g(t):
The domain of \( f(t) + g(t) \) is determined by the valid inputs for \( t \):
- The domain of \( f(t) \) is typically the time interval during which the rocket is in flight, which might start from \( t = 0 \) (launch time) and end at some positive \( t = T \) (when the rocket returns to the ground). Therefore, the time intervals may vary based on the specific rocket and launch conditions.
- Given that \( g(t) \) is constant (i.e., the platform height), it does not restrict the domain.
Thus, if we assume the rocket is launched at \( t = 0 \) and lands at \( t = T \), then the domain of \( f(t) + g(t) \) can be expressed as: \[ \text{Domain: } [0, T] \] where \( T \) is the total time of flight until the rocket hits the ground.
Summary
- \( h(t) = f(t) + g(t) \) represents the absolute height of the rocket above ground level over time.
- The domain for \( h(t) \) is the interval of time during which the rocket is in flight: \( [0, T] \), where \( T \) is the time when the rocket lands back on the ground.