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The problem involves analyzing the graph of \( f(x) = |x| \) and identifying the properties that apply. Let’s break it down:

### Key Features of \( f(x) = |x| \):
1. **Graph Shape**: The graph of \( f(x) = |x| \) is a "V" shape with its vertex at \( (0, 0) \). It is symmetric about the y-axis.
2. **Increasing/Decreasing Behavior**:
   - \( f(x) \) decreases on the interval \( (-\infty, 0] \).
   - \( f(x) \) increases on the interval \( [0, \infty) \).
3. **Domain and Range**:
   - Domain: All real numbers (\( (-\infty, \infty) \)).
   - Range: Non-negative real numbers (\( [0, \infty) \)).
4. **Even/Odd Function**:
   - \( f(x) \) is an **even function** because \( f(-x) = f(x) \) for all \( x \).
5. **Linear Function**:
   - \( f(x) \) is not linear overall, as it is piecewise linear (two linear parts joined at the vertex).

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### Properties to Select:
From the provided options:
- **A. The function is increasing on the interval \( (-\infty, \infty) \):** Incorrect. \( f(x) \) increases only on \( [0, \infty) \).
- **B. The function \( f \) is an even function:** Correct. \( f(x) = |x| \) is even.
- **C. The domain of \( f \) is \( (-\infty, \infty) \):** Correct.
- **D. The function \( f \) is an odd function:** Incorrect. \( f(x) \) is even, not odd.
- **E. The range of \( f \) is \( (-\infty, \infty) \):** Incorrect. The range is \( [0, \infty) \).
- **F. The function \( f \) is a linear function:** Incorrect.
- **G. None of the above properties apply:** Incorrect, as some properties do apply.

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### Correct Selections:
- **B. The function \( f \) is an even function.**
- **C. The domain of \( f \) is \( (-\infty, \infty) \).**

Would you like a more detailed explanation of any part?  

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### Related Questions:
1. What is the difference between even and odd functions?
2. How does the range of \( f(x) = |x| \) compare to its domain?
3. Can a function be both even and linear? Why or why not?
4. How do transformations affect the graph of \( f(x) = |x| \)?
5. What are other examples of piecewise functions, and how do their domains and ranges work?

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**Tip**: To confirm if a function is even or odd, substitute \( -x \) into \( f(x) \). If \( f(-x) = f(x) \), it's even; if \( f(-x) = -f(x) \), it's odd.

Sketch the graph of the following function and identify all properties that apply. f(x) = |x|. Identify the properties: Select all that apply.

Analyze the graph of f(x) = |x| to identify its properties, including domain, range, and symmetry. Explore key mathematical concepts like even functions and graph transformations, suitable for Grades 9-11.

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