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Understanding the domain of a function is a crucial step in analyzing any graph. For the logarithmic function \( y = 5 \log_{10}|x| \), it's important to note that the logarithm is only defined for positive numbers. This is because you can't take a logarithm of zero or a negative number and expect a real number result.

In this function, the use of the absolute value \( |x| \) ensures that we'll only be dealing with non-negative values of \( x \) because \( |x| \) transforms any negative input into a positive one. As a result, the domain of \( x \) is every real number except zero, which can be mathematically written as \( x \in (-\infty, 0) \cup (0, \infty) \).

This statement means you can use any positive or negative number for \( x \), just not zero itself. Zero is not included because \( \log_{10}(0) \) is undefined, as the logarithm of zero tends towards negative infinity.

Using a graphing calculator allows for an accurate visual representation of mathematical functions. When working with the function \( y = 5 \log_{10}|x| \), it is essential to make sure your calculator is set up correctly to handle both logarithmic functions and absolute values.

Here are the steps to graph this function on a graphing calculator:

• Start by entering the function into the calculator as it's written: \( y = 5 \log_{10}(|x|) \).
• Double-check that your calculator is in the appropriate mode for logarithms and absolute values.
• Ensure the domain is set correctly to exclude \( x = 0 \). This can often be done by setting exclusions in your calculator.

The graph of this function will show two symmetrical branches, explaining how the absolute value impacts the graph, allowing both positive and negative values of \( x \) to be used without violating the restrictions of the logarithmic function.

The concept of absolute value is vital when working with the function \( y = 5 \log_{10}|x| \). The absolute value is a mathematical function that takes a number and makes it non-negative, effectively removing any negative sign.

Mathematically, this is expressed as:

• If \( x \geq 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \), which converts \( x \) into its positive form.

In this function, the absolute value ensures that \( x \) is valid for the logarithm by making it positive or leaving it positive if it was already. This feature of \( |x| \) allows the function to handle both positive and negative values of \( x \), which is why the graph appears mirror-like, with symmetrical branches extending from both positive and negative directions. This symmetry results from keeping the input values for the logarithm non-negative, regardless of whether \( x \) starts off negative or positive.