Math Problem Solver

The domain of \(6 - \frac{1}{x} + \frac{1}{x}\) is \(\left(-\infty, 0\right) \cup \left(0, \infty\right)\).

\(6 - \frac{1}{x} + \frac{1}{x}\) has a removable singularity at \(x = 0\). A removable singularity of a function \(f\) is a point \(p\) at which \(f\) is undefined and where it is possible to redefine \(f\) such that the resulting function is continuous at \(p\). The resulting function is known as the continuous extension of \(f\) to \(p\).

The continuous extension of \(6 - \frac{1}{x} + \frac{1}{x}\) to \(0\) is:\[F(x) = \begin{cases} 6 & \text{for}\: x = 0 \\6 - \frac{1}{x} + \frac{1}{x} & \text{otherwise} \end{cases}\] The domain of \(F(x)\) is \(\left(-\infty, \infty\right)\).