Final answer:
The domain of the function n(t) is the set of all real numbers from 0 to infinity. It has specific values based on the range of t. For t less than or equal to 25, it is f(t). For t between 25 and 150, it is 200. For t greater than or equal to 150, it is 80t^2. And for t greater than or equal to 200, it is 0.05t.
Step-by-step explanation:
The domain of the function n(t) is the set of all real numbers from 0 to infinity. This means that any real number greater than or equal to 0 can be plugged into the function. However, there are some specific values for t where n(t) takes on different forms. For t less than or equal to 25, n(t) is equal to f(t). For t between 25 and 150, n(t) is equal to 200. For t greater than or equal to 150, n(t) is equal to 80t^2. Finally, for t greater than or equal to 200, n(t) is equal to 0.05t.
In the given expression for the function n(t), there are multiple cases or pieces defined for different intervals of t. Let's break down each piece:
f(t) for 25t ≤ t < 150:
In this interval, the function n(t) takes on the value of another function f(t) for values of t between 25t and 150.
It's important to note that unless specified, the domain of f(t) is assumed to be the set of all real numbers for which f(t) is real. So, for this interval, it's assumed that t can take any real value between 25t and 150.
200 for 150 ≤ t < 200:
For values of t between 150 and 200, the function n(t) is a constant value of 200.
80t^2 for 0 ≤ t < 0.05t:
In this interval, which is from 0 to 0.05t, the function n(t) is defined as 80t^2.
Similarly, it's assumed that t can take any real value between 0 and 0.05t for this piece.
Now, to summarize, the function n(t) has different rules or pieces for its behavior in different intervals of t. The domain for each interval is implied based on the given expressions and intervals, and it's typically assumed to be the set of real numbers for which the function is defined.
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