The function: f(x)=3x^4 - 2x^3 + 2x -1 is a polynomial function. Polynomial functions have several rules, characteristics, and definitions that are associated with them. For this particular polynomial function, the following rules are applicable:
1. Degree of the Polynomial: The degree of a polynomial is determined by the highest power of the variable x that appears in the polynomial. For our function f(x), the degree of the polynomial is 4, as identified by the highest exponent of x which is 4 in the term 3x^4.
2. Polynomial Coefficients: The coefficients of a polynomial are the numbers that are multiplied by the variable x. In our function f(x), the coefficients are 3, -2, 2, and -1, which are associated with the terms x^4, x^3, x, and the constant term respectively.
3. Polynomial Rules: The function is a polynomial, it means that it must follow the rules/pattern of a Polynomial function. These can include specific characteristics such as: the function is smooth and continuous, it should have no sharp corners or discontinuities, the end behavior should resemble either positive/negative infinity as x approaches positive/negative infinity, the polynomial should turn at most n-1 times where n is the degree of the polynomial (3 times in this case as the degree is 4). However, checking these features might require more specific information or graphical representation.
4. Legality of operation: In a polynomial, we can add, subtract, and multiply terms together, and we get another polynomial. Just one thing that is not allowed is dividing by a variable (for example, 2/(x) is not a polynomial).
5. Constant term: In the polynomial f(x), -1 is a constant term.
These are the rules that define the structure and characteristics of the given function, f(x).