An increasing function, denoted as f(x), is a mathematical function where the output value increases as the input value increases. This concept is fundamental in calculus and is characterized by a positive slope on the function's graph. When f(x) is an increasing function, it implies that for any two points x1 and x2 in the domain, where x1 < x2, the following inequality holds: f(x1) ≤ f(x2).

The reason why increasing functions are important is that they represent real-world scenarios where a quantity or value increases as another quantity increases. For instance, in economics, the demand for a product often increases as the price decreases, leading to an increasing demand function. In physics, the velocity of an object increases as the time elapsed increases, resulting in an increasing velocity function.

Data supporting the significance of increasing functions can be found in various fields. In economics, the demand curve typically slopes downwards, indicating an inverse relationship between price and quantity demanded. This is a decreasing function, but the concept of increasing functions is crucial in understanding the relationship between price and other economic variables, such as income or advertising expenditure.

In physics, the velocity of an object moving in a straight line is an increasing function of time, assuming constant acceleration. The velocity-time graph for such an object would be a straight line with a positive slope, illustrating the increasing nature of the function.

In summary, the concept of an increasing function is essential in mathematics and its applications across various disciplines. It represents a fundamental relationship where the output increases with the input, and it is supported by empirical data from economics and physics.