Final answer:
Option b. y = x² could correspond to f(x), as it exhibits a positive slope that decreases in magnitude with increasing x, which aligns with the function's described behavior at x = 3. The range of a horizontal line described as f(x) = a is y .
Step-by-step explanation:
To determine which option could correspond to the function f(x) given the information that at x = 3, f(x) has a positive value with a positive slope that is decreasing in magnitude with increasing x, we look at the provided options:
- a. y = 13x: This is a linear function with a positive slope that does not decrease in magnitude as x increases; the slope remains constant.
- b. y = x²: This is a quadratic function where the slope, given by the derivative 2x, is positive when x is positive and decreases in magnitude as x increases beyond 1.
Thus, given the description of the function's behavior at x = 3, option b. y = x² could correspond to f(x) since its slope is positive and decreasing in magnitude as x increases.
Regarding the range of a function that is a horizontal line, if the function’s equation is f(x) = a (where a is a constant), then for any value of x, the value of y (or f(x)) does not change and equals a. Therefore, the range of such a function is simply y = a.