The expression \((f + g)(t)\) represents the combined height of the rocket and the platform at any given time \(t\). Here, \(f(t)\) is the height of the rocket above the ground after \(t\) seconds, and \(g(t)\) is a constant height representing the height of a stationary platform, which does not change with time.
Description of the Situation:
- Rocket Height: \(f(t)\) represents the vertical height of a rocket as it ascends from the ground. As time progresses, the rocket gains altitude until it decreases back to the ground (due to gravity or controlled descent).
- Platform Height: \(g(t)\) is a constant height of a platform, suggesting that it is at a fixed point (for instance, a launch pad or any structure that is stationary at a certain height above the ground).
- Combined Height: The function \((f + g)(t)\) essentially gives the total height from the ground up to the rocket's height at time \(t\) plus the height of the platform. This could represent scenarios such as:
- A person standing on the platform trying to measure how high they are above the ground while simultaneously looking at the altitude of a rocket that was launched from that platform.
Domain of \((f + g)(t)\):
- The domain of \((f + g)(t)\) is determined by the overlapping domains of \(f(t)\) and \(g(t)\).
- \(f(t)\) is typically valid for \(t \geq 0\) since time cannot be negative. Thus, it represents the time starting from the moment of launch.
- Since \(g(t)\) is a constant function, it doesn't impose any additional restrictions on the domain.
- Therefore, the domain of \((f + g)(t)\) can be expressed as:
\[ \text{Domain}: {t \mid t \geq 0 } \]
This reflects all non-negative real numbers starting from the moment the rocket is launched.