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Functions describe situations where one quantity determines another. For example, the return on $10,000 invested at an annualized percentage rate of 4.25% is a function of the length of time the money is invested. Because we continually make theories about dependencies between quantities in nature and society, functions are important tools in the construction of mathematical models.

A function can be described in various ways, such as by a graph, e.g., the trace of a seismograph; by a verbal rule, as in, “I’ll give you a state, you give me the capital city;” by an algebraic expression like 𝑓(𝑥) = 𝑎 + 𝑏𝑥; or by a recursive rule. The graph of a function is often a useful way of visualizing the relationship of the function models, and manipulating a mathematical expression for a function can throw light on the function’s properties.

In school mathematics, functions usually have numerical inputs and outputs and are often defined by an algebraic expression. For example, the time in hours it takes for a car to drive 100 miles is a function of the car’s speed in miles per hour, 𝑣; the rule 𝑇(𝑣) = 100/𝑣 expresses this relationship algebraically and defines a function whose name is 𝑇.

Functions presented as expressions can model many important phenomena. Two important families of functions characterized by laws of growth are linear functions, which grow at a constant rate, and exponential functions, which grow at a constant percent rate. Linear functions with a constant term of zero describe proportional relationships.