Question
A rocket is shot up into the air. If f(t) represents the height in feet of the rocket at time, t, seconds and g(t) is a constant function that represents the height of a platform, what situation can be described by (f +g)(t)? Include a description of the domain of (f +g)(t)
A: The function (f +g)(t) is the height in inches at time, t, seconds of the rocket landed on the platform. The domain of the function is the set of all real numbers
A: The function (f +g)(t) is the height in inches at time, t, seconds of the rocket landed on the platform. The domain of the function is the set of all real numbers
Answers
GPT-4o mini
The expression \((f + g)(t)\) represents the combined height of the rocket and the height of the platform at a given time \(t\). Here, \(f(t)\) is the height of the rocket at time \(t\) seconds, and \(g(t)\) is a constant height that represents the height of the platform above ground level.
When you add these two functions, \((f + g)(t)\) indicates the total height above ground of the rocket at any given time \(t\), assuming the rocket is above the platform. Specifically, this could describe a scenario where the rocket has landed or is hovering directly above the platform, and thus the combined height is the height of the rocket plus the height of the platform.
### Description of the Domain:
The domain of the function \((f + g)(t)\) includes all values of \(t\) for which \(f(t)\) is defined and meaningful. In practical terms, this typically means:
1. **Pre-launch**: The time before the rocket is launched (e.g., \(t = 0\)).
2. **During flight**: The time when the rocket is in the air and moving up or down.
3. **Landing**: The time when the rocket lands back on the platform.
For most scenarios, you would specify the domain as \(t \geq 0\) to reflect the non-negative time after launch. However, in a broader sense, if you're considering all possible times (including potentially negative or non-physical scenarios), and if \(f(t)\) is defined for all real numbers (which is less common), then the domain of \((f + g)(t)\) could indeed be described as the set of all real numbers \((-\infty, \infty)\).
### Summary
In summary, \((f + g)(t)\) describes the combined height of the rocket above the platform at time \(t\), and the domain can be detailed based on the context of the rocket's flight, commonly set as \(t \geq 0\) for practical scenarios.
When you add these two functions, \((f + g)(t)\) indicates the total height above ground of the rocket at any given time \(t\), assuming the rocket is above the platform. Specifically, this could describe a scenario where the rocket has landed or is hovering directly above the platform, and thus the combined height is the height of the rocket plus the height of the platform.
### Description of the Domain:
The domain of the function \((f + g)(t)\) includes all values of \(t\) for which \(f(t)\) is defined and meaningful. In practical terms, this typically means:
1. **Pre-launch**: The time before the rocket is launched (e.g., \(t = 0\)).
2. **During flight**: The time when the rocket is in the air and moving up or down.
3. **Landing**: The time when the rocket lands back on the platform.
For most scenarios, you would specify the domain as \(t \geq 0\) to reflect the non-negative time after launch. However, in a broader sense, if you're considering all possible times (including potentially negative or non-physical scenarios), and if \(f(t)\) is defined for all real numbers (which is less common), then the domain of \((f + g)(t)\) could indeed be described as the set of all real numbers \((-\infty, \infty)\).
### Summary
In summary, \((f + g)(t)\) describes the combined height of the rocket above the platform at time \(t\), and the domain can be detailed based on the context of the rocket's flight, commonly set as \(t \geq 0\) for practical scenarios.