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Four different functions are bounded by certain values along the $x$-axis. What I want to know is if there is one function that can describe all points in the set.

To be more specific, I have four bounds: $[1, 50]$, $[51, 68]$, $[69, 98]$, $[99, 100]$. Between each of the bounds, one specific function is used to determine the value (i.e. all values of $1 \le x \le 50$ are calculated as $y=x$, for all values $51 \le x\le 68$, the value is $y=x^2$, etc.).

My question is, how would I go about determining an equation to fit all of the points? I asked a buddy of mine, and he said that I would need a 99th degree polynomial, but that seems very wrong to me.

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It seems that the best way is to do nothing, that is to use indicator functions of the segments. For instance, put $$f(x)={\mathbf 1}_{[1,50]} x+\mathbf 1_{[51,68]} x^2+\dots.$$

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