In general, a function approximation problem asks us to select a function among a well-defined class^{[citation needed][clarification needed]} that closely matches ("approximates") a target function^{[citation needed]} in a task-specific way.^{[1][better source needed]} The need for function approximations arises in many branches of applied mathematics, and computer science in particular^[why?],^{[citation needed]} such as predicting the growth of microbes in microbiology.^[2] Function approximations are used where theoretical models are unavailable or hard to compute.^[2]

One can distinguish^{[citation needed]} two major classes of function approximation problems:

First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).^[3]

Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points of the form (x, g(x)) is provided.^{[citation needed]} Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification problem instead.^[4]

To some extent, the different problems (regression, classification, fitness approximation) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems.^{[citation needed]}