The question makes no sense as written. If you are asking for a function, f, that will give f(0)= 1/2, f(1)= 1/12, f(2)= 1/12, f(3)= 19/120, etc., then there are an infinite number of such functions. It is true that there exists a unique nth degree polynomial that will fit n+ 1 data points so, since you have 10 data points, there exist a unique 9th degree polynomial that will fit them. One way to find that polynomial is to write it as \(\displaystyle y= ax^9+ bx^8+ cx^7+ dx^6+ ex^5+ fx^4+ gx^3+ hx^2+ ix+ j\). Putting the given points for x and y gives 10 equations to solve for the 10 coefficients. You can also use "Newton's divided difference formula" https://www.geeksforgeeks.org/newtons-divided-difference-interpolation-formula/ to find that polynomial.
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