Basic requirement for interpolation & extrapolation to work, Mathematics

What is the basic requirement for both interpolation and extrapolation to work? There must exist a functional relationship between an independent variable and a dependent variable and it must be required to estimate the value of the dependent variable corresponding to a certain value of independent variable.  This functional relationship is usually developed on the basis of past data using a branch of Statistics is called Regression Analysis.

In the situations considered above, population is taken to be a function of time or years. While time is the independent variable, population is the dependent variable. One more point that requires emphasis is that both interpolation and extrapolation provides us with only estimates of the dependent variable.

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