Well, darling, the notation that shows P is a function of x is P(x). Yep, it's as simple as that. Just slap those two together with parentheses and you've got yourself a function relationship. Easy peasy lemon squeezy!
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It is useful to know the linear factors of a polynomial because they give you the zeros of the polynomial. If (x-c) is one of the linear factors of a polynomial, then p(c)=0. Here the notation p(x) is used to denoted a polynomial function at p(c) means the value of that function when evaluated at c. Conversely, if d is a zero of the polynomial, then (x-d) is a factor.
The normal way of showing that y is a function of x is to write y = f(x).
If you mean: x-2y = 8 then it is y = 0.5x-4.
The function in algebra of ordered pairs is function notation. For example, it would be written out like: f(x)=3x/4 if you wanted to know three fourths of a number.
To evaluate a function means to replace the variable with some value, and calculate the value of the function. For example, in the parabola y = x2 (or, using functional notation, f(x) = x2), if you replace x with 10, and calculate x2, you are evaluating the function for that specific value.