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The function given is:

\[
f(x) = |x - 2| - 8
\]

To find the domain of this function, we analyze the expression inside the function.

### Step 1: Identifying Restrictions
The expression involves an absolute value, \( |x - 2| \), and a constant \( -8 \). The absolute value function is defined for all real numbers, meaning that there are **no restrictions** on the values that \( x \) can take.

### Step 2: Domain of the Function
Since there are no restrictions, the domain of \( f(x) \) includes all real numbers. In interval notation, the domain is:

\[
\boxed{(-\infty, \infty)}
\]

If you want to confirm using graphing technology, you can graph \( f(x) = |x - 2| - 8 \) and observe that the graph is continuous for all \( x \), without breaks or undefined points.

Let me know if you need further explanation on graphing this function!

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### Related Questions:
1. What is the range of the function \( f(x) = |x - 2| - 8 \)?
2. How does shifting affect the graph of \( f(x) = |x| \)?
3. Can absolute value functions ever be undefined?
4. What does the graph of \( f(x) = |x - 2| - 8 \) look like?
5. How do transformations affect the shape of absolute value graphs?

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**Tip:** Absolute value functions are always defined for all real numbers because they output non-negative values regardless of the input.

Use graphing technology to find the domain of the function f(x) = |x - 2| - 8

Learn how to find the domain of the function f(x) = |x - 2| - 8 using graphing technology. This algebraic problem explores the concept of absolute value functions and explains how the domain of such functions includes all real numbers. Suitable for students in Grades 9-11.

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