The function \(f(t)\) represents the height of a rocket shot straight up, and \(g(t)\) is a constant function representing the height of a platform. The expression \((f + 9)(t)\) can be interpreted as the height of the rocket at time \(t\), increased by 9 feet. This could represent a scenario where the rocket is positioned 9 feet above the original height defined by \(f(t)\).
For example, if the rocket launches from a height of \(f(t)\) feet above ground level, then \((f + 9)(t)\) would represent the height of the rocket plus an additional 9 feet, suggesting that the rocket is either launched from a platform that is itself 9 feet high or that it is simply at an elevation of 9 feet above its original height at any given moment in time \(t\).
Domain of \((f + g)(t)\)
The domain of \((f + g)(t)\) consists of all real numbers \(t\) where both functions \(f(t)\) and \(g(t)\) are defined. Since \(g(t)\) is a constant function, it is defined for all \(t\). The function \(f(t)\), representing the rocket's height, is also defined for a specific time interval — typically starting from \(t = 0\) (the launch time) and continuing until the rocket either hits the ground or reaches a maximum height and begins to descend.
Thus, the domain of \((f + g)(t)\) would generally be expressed in terms of the time interval from launch until the rocket lands back on the ground. This could be something like \([0, T]\), where \(T\) is the time the rocket takes to return to the ground. If \(f(t)\) is defined on \([0, T]\), then \((f + g)(t)\) would have the same domain: \([0, T]\).
In summary, \((f + 9)(t)\) reflects the height of the rocket at time \(t\) plus an additional 9 feet, and the domain of \((f + g)(t)\) is determined by the interval where the rocket's height is defined.