Find the domain of the function:
f ( x )

The function given is:

$f(x) = \frac{1}{2x - 2}$

To find the domain of the function:

The denominator of the function, $2x - 2$, cannot be zero because division by zero is undefined. So, let's set the denominator equal to zero and solve for $x$:

$2x - 2 = 0$

Solve for $x$:

$2x = 2$ $x = 1$

So, $x = 1$ would make the denominator zero, and the function is undefined at $x = 1$.

Therefore, the domain of the function is all real numbers except $x = 1$:

$(-\infty, 1) \cup (1, \infty)$

Looking at the options, the correct answer is:

$(-\infty, 2) \cup (2, \infty)$

(Note that this interval also avoids the point $x = 1$, where the function is undefined.)

Would you like further clarification on this? Here are some additional questions to help explore the concept:

1. What happens to the function as $x$ approaches $1$?
2. How can we identify the domain of rational functions in general?
3. What is the significance of excluding certain values from the domain in terms of real-world applications?
4. Can a rational function ever have more than one point of discontinuity?
5. How would you handle the domain of a function involving square roots or logarithms?

Tip: Always check for values that make the denominator zero or any values that would cause other mathematical operations (like square roots or logarithms) to be undefined when finding the domain.