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The function given is \( f(x) = 3|x| \), which represents the absolute value function scaled by 3. Let's analyze whether this function is even or odd.

1. **Even Function Test:**
   A function is even if \( f(-x) = f(x) \) for all values of \( x \).

   - For \( x = -2 \), \( f(-2) = 3|{-2}| = 6 \) and \( f(2) = 3|2| = 6 \).
   - For \( x = -1 \), \( f(-1) = 3|{-1}| = 3 \) and \( f(1) = 3|1| = 3 \).
   - For \( x = 0 \), \( f(0) = 0 \) (which is the same for both \( x = 0 \) and \( -x = 0 \)).

   Since \( f(-x) = f(x) \) for all values in the table, **the function is even**.

2. **Odd Function Test:**
   A function is odd if \( f(-x) = -f(x) \) for all values of \( x \). This does not hold true because \( f(-2) = 6 \) but \( -f(2) = -6 \), and similarly for other values, so the function is not odd.

**Conclusion:** The function \( f(x) = 3|x| \) is **even**.

Would you like more details on how to determine whether a function is even or odd? Here are some related questions:

1. What are the characteristics of even and odd functions in general?
2. How does the absolute value function affect the evenness of a function?
3. Can a function be neither even nor odd? If so, how can you tell?
4. What are some examples of odd functions, and how do they differ from even functions?
5. How would the graph of \( f(x) = 3|x| \) look visually?

**Tip:** Even functions are symmetric about the y-axis, while odd functions are symmetric about the origin.

Learn how to determine if the function f(x) = 3|x| is even or odd with a step-by-step explanation. This example covers key algebraic concepts, such as the absolute value function and symmetry tests, and is suitable for students in Grades 9-11.

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