A monotone function is always

A monotone function is always

• A. The function is always increasing or always decreasing
• B. The function is constant
• C. The function is periodic
• D. The function is sometimes increasing and sometimes decreasing

Answer: (A)

A monotone function keeps the same direction of change across its entire domain: as x increases, the function values never switch from increasing to decreasing (or vice versa). In other words, it either never goes down (nondecreasing) or never goes up (nonincreasing) as you move along x. That means the function is always moving in one direction: increasing (or staying flat) or decreasing (or staying flat) throughout its domain. That’s why the statement that it is always increasing or always decreasing is the best description. A constant function is a special case that fits monotonicity, but not every monotone function is strictly increasing or strictly decreasing. A function that is periodic would have to repeat values in cycles, which conflicts with the idea of not reversing direction. And a function that is sometimes increasing and sometimes decreasing would violate monotonicity by changing direction.

A monotone function keeps the same direction of change across its entire domain: as x increases, the function values never switch from increasing to decreasing (or vice versa). In other words, it either never goes down (nondecreasing) or never goes up (nonincreasing) as you move along x. That means the function is always moving in one direction: increasing (or staying flat) or decreasing (or staying flat) throughout its domain.

That’s why the statement that it is always increasing or always decreasing is the best description. A constant function is a special case that fits monotonicity, but not every monotone function is strictly increasing or strictly decreasing. A function that is periodic would have to repeat values in cycles, which conflicts with the idea of not reversing direction. And a function that is sometimes increasing and sometimes decreasing would violate monotonicity by changing direction.