Decreasing Functions

Definition

A function $f$ is decreasing on an interval $I$ if $f(x_2) < f(x_1)$ whenever $x_1 < x_2$ for any two points $x_1, x_2$ in $I$.

• How to read: “The value f of x two is less than f of x one whenever x one is less than x two.”
• Meaning: Bigger input gives smaller output — the graph falls left to right on $I$.

Why It Matters

Understanding where a function is decreasing is essential for identifying inefficiencies, decay, or loss in a system. For example, in physics, it can describe cooling or radioactive decay; in finance, it can model depreciation or market downturns.

Core Concepts

• Monotonicity: A decreasing function is a type of monotonic function, maintaining a consistent downward trend.
• Derivative Test: If a function $f$ is differentiable on an open interval, and $f'(x) < 0$ for all $x$ in that interval, then $f$ is decreasing on that interval.
  • How to read: “If the derivative f prime of x is negative, the function is decreasing.”
  • Meaning: A negative slope means the function values are falling.
• Strictly Decreasing: If $f(x_2) < f(x_1)$ for $x_2 > x_1$, the function is strictly decreasing. If it is $f(x_2) \le f(x_1)$, it is non-increasing.

Connected Concepts

• Function Definition, Function Notation
• Function Domain
• Function Range
• Increasing Functions
• Graphs of Functions: Properties