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Understanding the properties of a function is crucial when determining if it is even, odd, or neither. This involves knowing how the function behaves for different inputs. In our exercise, the given function is \( f(x)=3 x^{6}+2 x^{2}+|x| \).

To identify whether it is even or odd, we need to understand the definitions and properties of even and odd functions. \textbf{Even Function:}
- A function, \( f(x) \), is even if \( f(-x) = f(x) \) for all x in its domain. This means that the function is symmetric with respect to the y-axis. Common examples include \( x^2 \), \( x^4 \), and in general, any function with all even-power terms.

\textbf{Odd Function:}
- A function, \( f(x) \), is odd if \( f(-x) = -f(x) \) for all x in its domain. This indicates symmetry about the origin. Examples include \( x \), \( x^3 \), and any function containing only odd-power terms.

By substituting \( -x \) into our function, we can compare \( f(-x) \) with \( f(x) \) and determine the function's properties.

In algebra, you often deal with manipulating expressions and substituting values to derive new forms of a function. Let's look again at \( f(x)=3 x^{6}+2 x^{2}+|x| \). We substituted \( -x \) to check the symmetry:

\textbf{Compute \( f(-x) \)}:
- Substitute \( -x \) into the function: \( f(-x) = 3(-x)^6 + 2(-x)^2 + |-x| \).

- Since \( (-x)^6 = x^6 \), \( (-x)^2 = x^2 \), and \( |-x| = |x| \):\( f(-x) = 3x^6 + 2x^2 + |x| \).

This result shows \( f(-x) \) matches \( f(x) \). Algebra concepts enable us to simplify and evaluate functions correctly through such substitutions, ensuring we understand their behavior.

Symmetry plays a critical role in determining if a function is even, odd, or neither. In our example, we found the function \( 3 x^{6}+2 x^{2}+|x| \) to be even because \( f(x) = f(-x) \).

For even functions, symmetry means that the graph of the function looks the same on the left and right sides of the y-axis. Each point \((x, y)\) has a corresponding point \((-x, y)\).

On the other hand, for odd functions, symmetry about the origin means each point \((x, y)\) has a corresponding point \((-x, -y)\). This is not the case with the function we evaluated.

Understanding symmetry helps you visualize the function's graph and its behavior under transformations. It’s an essential concept in function analysis and beyond.