Given that f(x)=6x2−2x and g(x)=2x, what is the domain of (fg)(x)

Question

Responses

the set of all real numbers except x=13

the set of all real numbers

the set of all real numbers except x=0 and x=13

the set of all real numbers except x=0

Answer 1

To determine the domain of the composite function \((fg)(x)\), we first need to understand the individual functions:

1. Function \(f(x) = 6x^2 - 2x\): This is a polynomial function, which is defined for all real numbers. Therefore, there are no restrictions on the domain from \(f(x)\).

2. Function \(g(x) = 2x\): This is also a polynomial function, which again is defined for all real numbers, so there are no restrictions on the domain from \(g(x)\) either.

The domain of the product \((fg)(x) = f(x) \cdot g(x)\) will be determined by the intersection of the domains of \(f(x)\) and \(g(x)\). Since both functions are defined for all real numbers, the product \(fg(x)\) is also defined for all real numbers.

Thus, the domain of \((fg)(x)\) is:

The set of all real numbers.

So the correct response is:

the set of all real numbers.