Decreasing Functions

A function y = f(x) is said to be decreasing on an interval I = (a, b) if, for any two points x₁ and x₂ within the interval such that x₁ < x₂, the following condition is satisfied: $$ f(x_1) \ge f(x_2) $$. It is called strictly decreasing if: $$ f(x_1) > f(x_2) $$

A strictly decreasing function is also described as decreasing in the strict sense.

Both decreasing and strictly decreasing functions belong to the broader class of monotonic functions.

Note: A function is monotonic on an interval if it is either consistently increasing or consistently decreasing throughout that interval.

Example

Consider the function:

$$ y = x^2 $$

This function is strictly decreasing on the interval (−4, 0), because for any two points x₁ and x₂ in that interval with x₁ < x₂, it holds that f(x₁) > f(x₂).

However, this function is not decreasing on the interval (0, 4).

Note: For continuous and differentiable functions, decreasing behavior can also be studied by analyzing the sign of the first derivative, f ′(x).

And so on.