The function is given as:

$f(x) = \frac{1}{\frac{10}{x - 2} - 2}$

To find the domain, we must determine for which values of $x$ the function is defined. The denominator of the function must not be zero, as division by zero is undefined.

Step 1: Look at the inner denominator $\frac{10}{x - 2}$.

The inner denominator is $x - 2$. We need to avoid making this zero:

$x - 2 \neq 0 \quad \Rightarrow \quad x \neq 2$

Step 2: Check the outer denominator.

Now, consider the outer denominator $\frac{10}{x - 2} - 2$. We need to ensure that this entire denominator is not zero, so:

$\frac{10}{x - 2} - 2 \neq 0$

Solving this inequality:

$\frac{10}{x - 2} \neq 2$

Multiply both sides by $x - 2$ (which we know is not zero):

$10 \neq 2(x - 2)$

Simplifying:

$10 \neq 2x - 4$

$2x \neq 14$

$x \neq 7$

Step 3: Combine results.

From the two conditions, we need to exclude $x = 2$ and $x = 7$ from the domain.

Domain:

Thus, the domain of $f(x)$ is all real numbers except $x = 2$ and $x = 7$. In interval notation, this is:

$(-\infty, 2) \cup (2, 7) \cup (7, \infty)$

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Would you like further details on any part of this explanation?

Here are some related questions:

1. What happens if we substitute $x = 2$ or $x = 7$ into the original function?
2. How can we check if a function is defined for certain values of $x$?
3. What are the types of discontinuities that could occur in rational functions?
4. How would you find the domain of a function that involves square roots?
5. How does the domain of a function relate to the behavior of its graph?

Tip: When determining the domain, always look out for values that would make any denominator equal to zero or cause other undefined operations like taking the square root of negative numbers.