Final answer:
The question involves analyzing a function and a piecewise function with domain considerations, inverse variations, and probability distributions within the given conditions, with an emphasis on theoretical and applied mathematics.
Step-by-step explanation:
The question presented deals with the analysis of functions, specifically with the function f(x) = 1/x and a piecewise function g(x) that varies depending on the sign of x. When considering the function f(x) = 1/x, for the domain 0 ≤ x ≤ 20, we have a hyperbolic function that is defined only for x > 0, as division by zero is undefined. Because this is also a continuous function, for any x within the given domain (excluding zero), f(x) will yield a corresponding y-value where the product of f(x) and x is always 1.
The idea presented here is that as x increases, f(x) decreases inversely since their product is constant, a concept known as inverse variation. Additionally, there is a reference to probability functions and normal distribution, indicating a broader context of statistical analysis, possibly about probability distribution functions. For instance, when analyzing a continuous probability function f(x) with a range restriction, one might want to find the probability P(0 < x < 12) by integrating the function over the stated interval.
When evaluating expressions with both positive and negative signs in the numerator, it's typical to check both resulting values of x to see which is feasible within the context of a specific problem, often discarding the one that makes no sense for the given scenario, in this case possibly regarding chemical concentrations.