Final answer:
The domain and codomain of the function f(x) = 3x² - 6x + 3 are both all real numbers, since it is a polynomial which is defined for all real inputs and is structurally capable of yielding any real number output.
Step-by-step explanation:
To find the domain and codomain of f(x) = 3x² - 6x + 3, we must understand the nature of the function. This function is a polynomial of degree 2, which means its graph is a parabola. Since polynomials are defined for all real numbers, the domain of this function is all real numbers. This is represented in interval notation as –[∞, +∞] (Option b).
The codomain of a function is the set of all possible values that f(x) can take. For a quadratic function with a positive leading coefficient (such as 3 in this case), the graph opens upwards and there is a minimum value that the function can take. However, since the question only asks for the codomain and not the range, and there are no restrictions described in the function's expression, the codomain is also all real numbers, since the function is capable of yielding any real number given some input in the domain. Therefore, the correct option is: Domain: All real numbers; Codomain: All real numbers (Option a).