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In general, a function approximation problem asks us to select a function that closely matches ("approximates") a function in a task-specific way. The need for function approximations arises, for example, predicting the growth of microbes in microbiology. Function approximations are used where theoretical models are unavailable or hard to compute.
First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, ) can be approximated by a specific class of functions (for example, or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).
Secondly, for example, if g is an operation on the , 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.
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