Final answer:
A decreasing function decreases as x increases, and the inverse function will also be decreasing. For the original function described, option b. y = x² is correct as the slope is positive and decreases in magnitude. In the relationship with a negative slope, such as altitude and air density, y decreases as x increases.
Step-by-step explanation:
A decreasing function satisfies the condition where the function's values decrease as the input (usually represented as x) increases. Therefore, for any two points a and b on the function where a < b, f(a) > f(b).
Regarding the inverse of such a function, two statements must be true. First, if the function is strictly decreasing, then the inverse function will also be strictly decreasing. This is because the roles of x and y are essentially swapped in the inverse. Second, if the original function has a negative slope, as indicated by the decrease of y when x increases, then the inverse function will have a positive slope. Imagine flipping the graph over the line y = x; a downward trend becomes an upward trend.
The student's question suggests analyzing different descriptions of functions and determining which could indicate a decreasing function or its inverse. Given the provided information, a function with a positive value at x = 3 and a positive slope that is decreasing in magnitude with increasing x aligns with option b. y = x², as the slope (2x) is positive and decreases as x approaches zero. In the context of slopes and relationships, if we consider the altitude-air density relationship that has a negative slope, similar principles apply—the higher the altitude, the lower the air density.