Final answer:
The product of the functions f(x) = 3x and g(x) = x² - 1 is (fg)(x) = 3x³ - 3x. The domain of both f(x) and g(x) individually, and thus of their product (fg)(x), is all real numbers.
Step-by-step explanation:
To find the product of the functions f(x) and g(x), identified as (fg)(x), we must multiply f(x) by g(x). In this case, f(x) = 3x and g(x) = x² - 1. Therefore, we have:
(fg)(x) = f(x) × g(x) = (3x) × (x² - 1) = 3x³ - 3x
The domain of the new function, (fg)(x), is the set of all real numbers for which the function is defined. Since f(x) is a linear function and g(x) is a quadratic function, both are defined for all real numbers. Thus, the domain of (fg)(x) is all real numbers (x such that x is a real number).
Domain of f(x): All real numbers.
Domain of g(x): All real numbers.
By considering the individual domains of f(x) and g(x), we conclude that the domain of the product (fg)(x) remains the same as the domain of each function separately, which is all real numbers.