1. **Find the domain of the function** $f(x) = \frac{1}{x - 1}$. 2. The domain of a function is the set of all $x$ values for which the function is defined. 3. For $f(x) = \frac{1}{x - 1}$, the denominator cannot be zero because division by zero is undefined. 4. Set the denominator equal to zero and solve: $$x - 1 = 0$$ $$x = 1$$ 5. So, $x = 1$ is excluded from the domain. 6. Therefore, the domain is all real numbers except $1$, which in interval notation is: $$(-\infty, 1) \cup (1, \infty)$$