Increasing Function

An increasing function exhibits a rise in output values as input values increase. This behavior is depicted graphically as a curve or line that slopes upward from left to right.

Such a function satisfies the condition that if x₁ < x₂, then f(x₁) < f(x₂), indicating that the function values grow with increasing inputs. This concept is fundamental in understanding growth trends across various domains, such as population dynamics, financial investments, or resource consumption.

The average rate of change of a function over a specific interval measures how quickly the function’s output changes relative to its input. It is computed using the formula:

where a and b are two distinct input values and f(a) and f(b) are the corresponding outputs. This calculation yields a single value representing the function’s overall behavior across that interval, akin to finding the slope of a straight line

Geometrically, this rate corresponds to the slope of the secant line connecting the points (a, f(a)) and (b, f(b)) on the graph. If this line slopes upward, the function is increasing on the interval [a, b]. A positive average rate of change confirms the presence of growth during the period analyzed.

In real-world applications, identifying intervals where a function increases is essential. For instance, tracking the upward trend in a company’s revenue or the growth of a biological population over time requires analyzing such intervals. These methods support data-driven decision-making and help in modeling dynamic systems accurately.