Increasing Functions

Definition

A function $f$ is increasing 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 greater than f of x one whenever x one is less than x two.”
• Meaning: Bigger input gives bigger output — the graph rises left to right on $I$.

Why It Matters

Identifying where a function is increasing is crucial for optimization. In economics, it might represent a region of increasing returns; in engineering, it could indicate where a system’s output grows with input. It is a fundamental property used to locate local extrema and understand a function’s overall trend.

Core Concepts

• Monotonicity: An increasing function is a type of monotonic function, meaning it maintains a consistent direction of growth.
• 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 increasing on that interval.
  • How to read: “If the derivative f prime of x is positive, the function is increasing.”
  • Meaning: A positive slope means the function values are rising.
• Strictly Increasing: If $f(x_2) > f(x_1)$ for $x_2 > x_1$, the function is strictly increasing. If it is only $f(x_2) \ge f(x_1)$, it is non-decreasing.

Connected Concepts

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