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Math Help - Finding a specific funcion with parameters given, tough seeming question

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    Member mfetch22's Avatar
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    Finding a specific funcion with parameters given, tough seeming question

    Heres the question, I have tried to construct such a function, but I keep having trouble making it so both conditions are met, I seem to only be able to manufacture a function that works for one of the conditions, or the other, but not both simultaneous. I dont need a full blown answer, but I would definantly like some guidence as to where to go next:

    If x_1, ... , x_n are distinct numbers, find a polynomial function f_i of degree n-1 which is 1 at x_i and 0 at x_j for j \neq i.

    The question also gives a hint:

    The product of all (x-x_j) for j \neq i, is 0 at x_j if j \neq i.

    Thank you for any help on this question that anybody can give.
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    Quote Originally Posted by mfetch22 View Post
    Heres the question, I have tried to construct such a function, but I keep having trouble making it so both conditions are met, I seem to only be able to manufacture a function that works for one of the conditions, or the other, but not both simultaneous. I dont need a full blown answer, but I would definantly like some guidence as to where to go next:

    If x_1, ... , x_n are distinct numbers, find a polynomial function f_i of degree n-1 which is 1 at x_i and 0 at x_j for j \neq i.

    The question also gives a hint:

    The product of all (x-x_j) for j \neq i, is 0 at x_j if j \neq i.

    Thank you for any help on this question that anybody can give.
    Because it is a polynomial function of degree n-1 f_i(x)=A\prod_j(x-x_j).

    All that is left is to determine what A satisfies f_i(x_i)=1\therefore A=\dfrac{1}{\prod_j(x_i-x_j)}

    So finally you get f_i(x)=\dfrac{\prod_j(x-x_j)}{\prod_j(x_i-x_j)}
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