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Function approximation is the study of selecting functions in a class that match target functions. It’s a process that is useful in applied mathematics and computer science. Function approximation is often related to a Markov decision process (MDP) which consists of an agent and various states.
To understand function approximation well, it's important to know that in this term the word "function" doesn't refer to an object oriented programming function that takes a variable and provides a result. The word "function" refers to the mathematical use of function, where a function matches one item in a data set to another single item in another data set.
Another key point is that function approximation often works with value iteration in a MDP process. Mathematicians show how function approximation and value iteration can be used to build gameplay strategies for various video games, which is one of the most prominent and easiest ways to show how MDPs work.
In this and other kinds of predictive and modeling work based on MDPs, function approximation plays a key role.