Question

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

the set of all real numbers except x=13
the set of all real numbers except x is equal to 1 third

the set of all real numbers except x=0
the set of all real numbers except x is equal to 0

the set of all real numbers except x=0
and x=13
the set of all real numbers except x is equal to 0 and x is equal to 1 third

the set of all real numbers
the set of all real numbers

Answer 1

To find the domain of \((fg)(x) = f(x) \cdot g(x)\), we need to analyze the functions \(f(x)\) and \(g(x)\).

1. **Function \(f(x)\)**:
\[
f(x) = 6x^2 - 2x
\]
This is a polynomial function, and polynomial functions are defined for all real numbers. Thus, the domain of \(f(x)\) is all real numbers.

2. **Function \(g(x)\)**:
\[
g(x) = 2x
\]
This is also a polynomial function, which is defined for all real numbers. Thus, the domain of \(g(x)\) is all real numbers.

Now, to find \((fg)(x)\), we multiply the two functions together:
\[
(fg)(x) = f(x) \cdot g(x) = (6x^2 - 2x)(2x)
\]

Since both \(f(x)\) and \(g(x)\) are defined for all real numbers, \((fg)(x)\) will also be defined for all real numbers. Therefore, the domain of \((fg)(x)\) is the set of all real numbers.

So, the answer is:
**the set of all real numbers**.