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To understand the domain of a function, you need to know the set of all possible input values (x-values) for which the function is defined. For the logarithmic function \(f(x)=\log_{3}|x|\), the argument of the logarithm must be positive. This is because the logarithm of a negative number or zero is undefined in the set of real numbers.
Since \|x|\ (the absolute value of x) is always positive for non-zero x, the domain of \(f(x)=\log_{3}|x|\) is all real numbers except x = 0. Therefore, the domain is: \(x \in (-\infty, 0) \cup (0, \infty)\).
Understanding domain restrictions helps prevent errors in calculations and in interpreting graphs of functions.

The absolute value function, denoted \|x|\, measures how far a number is from zero on the number line, regardless of direction. Essentially, it converts any negative input into its positive equivalent. For example, \|3| = 3\ and \|-3| = 3\.
This property of absolute value is particularly useful in dealing with logarithmic functions like \(f(x)=\log_{3}|x|\). Despite \(x\) being negative, \|x|\ ensures the input to the logarithm remains positive. This helps in forming a function that is defined for all non-zero x.
The symmetry of absolute values also affects how the function looks on a graph. When graphing, values of \(f(x)\) remain the same for both positive and negative x, creating symmetry across the y-axis.

A graphing calculator can be invaluable for visualizing functions and their properties. To graph \(f(x)=\log_{3}|x|\) on a graphing calculator, follow these steps:

• Turn on the graphing calculator.
• Enter the function using the appropriate syntax: \(\log_{3}(|x|)\). Here, \(abs(x)\) denotes the absolute value.
• Set the viewing window to \(x: [-4, 4]\) and \(y: [-4, 4]\) to capture the desired range.
• Graph the function.

When interpreting the graph displayed by the calculator, be cautious. While the graph may suggest that x = 0 is part of the domain, remember that the absolute value ensures the argument of the logarithm is positive, thus excluding x = 0 from the domain. Always refer back to the calculated domain from earlier: \(x \in (-\infty, 0) \cup (0, \infty)\).