Before a derivative can be taken or an integral evaluated, an object must exist to differentiate or integrate: a function. In quantitative finance that object might be an option payoff, a yield curve, a probability density, or an implied volatility surface — each is a function, and each is described in one or more of four equivalent ways. This note fixes the formal definition, surveys the four representations, and introduces the catalogue of essential function families you will meet throughout the calculus sequence and in every pricing and risk model beyond it.
Learning objectives. By the end of this note you should be able to:
- state the precise definition of a function, its domain, codomain, and range;
- read off domain and range from a graph using the vertical and horizontal projections;
- move between verbal, numerical, visual, and algebraic representations of the same function;
- identify which of the six essential families (linear, polynomial, rational, power, exponential, trigonometric) a given function belongs to;
- construct a linear mathematical model from two data points and state its extrapolation risk.
In this lesson
- The definition
- The four representations
- The catalogue of essential functions
- Mathematical models and their limits
The full lesson, with worked examples and exercise solutions, is for members.
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