To determine the domain of the product of the functions \( f(x) \) and \( g(x) \), we first need to consider the domains of each function individually.
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Finding the domain of \( f(x) = 6x^2 - 2x \):
- This is a polynomial function, and polynomial functions are defined for all real numbers. Thus, the domain of \( f(x) \) is all real numbers: \[ \text{Domain of } f(x) = \mathbb{R} \]
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Finding the domain of \( g(x) = 2x \):
- This is also a polynomial function, which is defined for all real numbers. Therefore, the domain of \( g(x) \) is also: \[ \text{Domain of } g(x) = \mathbb{R} \]
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Finding the domain of the product \( (fg)(x) = f(x) \cdot g(x) \):
- The product of two functions is defined wherever both functions are defined. Since both \( f(x) \) and \( g(x) \) are defined for all real numbers, the domain of \( (fg)(x) \) is also: \[ \text{Domain of } (fg)(x) = \mathbb{R} \]
Since the product of \( f(x) \) and \( g(x) \) does not introduce any new restrictions, the domain of \( (fg)(x) \) is the set of all real numbers.
Final Answer: The set of all real numbers.