Decreasing Function

A decreasing function describes a relationship where the output consistently declines as the input increases. This means that for any two input values, if one is greater than the other, the corresponding output is smaller. Mathematically, a function f is decreasing on an interval I if for
every x₁ < x₂​ in I, f (x₁) > f (x₂). This type of behavior is visually identified on a graph that slopes downward from left to right.

The nature of a function can be analyzed by calculating its rate of change. For a function defined at discrete points, the average rate of change over an interval is the ratio of the change in the output to the change in the input:

If this value is negative across intervals, the function is decreasing. In continuous functions, the derivative f′(x) serves as an indicator—if f′(x) < 0 for all x in an interval, the function is decreasing on that interval.

Decreasing functions appear in many natural and technological contexts. Examples include the temperature of a cooling object, the voltage of a discharging battery, and the height of a falling object after its peak. These scenarios involve quantities that reduce as time or another input progresses, making decreasing functions essential for modeling and analyzing such phenomena.