Question

1. each of the functions p(t), b(h), and f(x) can be modeled by the same linear relationship.

discuss: why are the domains of these three functions different from each other?

Explanation:

The domain of a function depends on its real - world context (if it has one) or mathematical restrictions. For \(p(t)\), \(t\) might represent a quantity like time, which has non - negative values (e.g., time can't be negative). For \(b(h)\), \(h\) could represent height, which also has practical limits (e.g., height of an object can't be negative or might have an upper bound based on the situation). For \(f(x)\), if it's a purely mathematical function without a real - world context, \(x\) could be all real numbers. But when these functions model real - world phenomena, the variables \(t\), \(h\), and \(x\) represent different quantities with different practical constraints, leading to different domains even though the linear relationship (the rule of the function) is the same.

Answer:

The domains differ because the variables \(t\), \(h\), and \(x\) in \(p(t)\), \(b(h)\), and \(f(x)\) represent different quantities (e.g., time, height, a general variable) with different real - world or mathematical constraints. For example, \(t\) (time) and \(h\) (height) may have non - negative or bounded values in real - world models, while \(x\) in a pure math function might have a broader domain, and even in real - world contexts, different quantities have different valid value ranges.