1. The problem is to find the domain of the function $F(x) = 2x^2 + 1$. 2. The domain of a function is the set of all possible input values ($x$) for which the function is defined. 3. Since $F(x) = 2x^2 + 1$ is a polynomial function, it is defined for all real numbers. 4. Therefore, the domain of $F(x)$ is all real numbers, which can be written as: $$\text{Domain} = (-\infty, \infty)$$ 5. In simple terms, you can plug any real number into $x$ and get a valid output from $F(x)$. Final answer: The domain of $F(x) = 2x^2 + 1$ is all real numbers.