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Part 1
Find the domain of the function.
f left parenthesis x right parenthesis equals StartStartFraction 1 OverOver StartFraction 10 Over x minus 2 EndFraction minus 2 EndEndFraction
Question content area bottom
Part 1
The domain of​ f(x) is 
  
enter your response here.
​(Type your answer in interval​ notation.)

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:

\[
(-\infty, 2) \cup (2, 7) \cup (7, \infty)
\]

---

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.

Find the domain of the function f(x) = 1 / (10 / (x - 2) - 2). The domain of f(x) is ___. (Type your answer in interval notation.)

Learn how to find the domain of the function f(x) = 1 / (10 / (x - 2) - 2). This problem explores rational functions and determining where they are undefined. Key concepts include excluding values that make the denominator zero. Suitable for students in Grades 10-12.

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