91Ó°ÊÓ

For the function \( f(x) = \log_{2} |x| \), understanding the domain and range is crucial. The domain refers to the set of input values \( x \) for which the function is defined. In this case, because of the absolute value, \( |x| \) needs to be greater than zero. Thus, \( x \) cannot equal zero, making the domain \( x \in (-\infty, 0) \cup (0, \infty) \).
The range of a function refers to the set of possible output values. Here, \( \log_{2} |x| \) allows for all real values of outputs since \(|x|\) can take any positive value, leading the logarithm to potentially yield all real numbers. Hence, the range is \((-\infty, \infty)\).
This comprehensive view of domain and range is vital for graphing the function and understanding its behavior.

Exploring the symmetry of a function provides insights into its graph. The symmetry simplifies graphing as it reduces the amount of information we need to determine the full graph. The function \( f(x) = \log_{2} |x| \) exhibits even symmetry.
Even functions satisfy the condition \( f(-x) = f(x) \). Let's verify this for \( \log_{2} |x| \):

• For any input \( x \), \( f(-x) = \log_{2} |-x| \).
• Since \( |-x| = |x| \), then \( \log_{2} |-x| = \log_{2} |x| \).

Thus, indeed \( f(-x) = f(x) \), confirming the function is even. This implies the graph is symmetric with respect to the y-axis, meaning for every point \( (x, y) \) on the graph, there is a corresponding point \( (-x, y) \). Appreciating symmetry helps in quick visualization and sketching of functions.

Graphing \( f(x) = \log_{2} |x| \) involves understanding the function's characteristics and key points.
First, plot essential points on the graph, which give a clear structure to the whole image. Given its symmetry, focus on some key positive \( x \)-values:

• At \( x = 1 \), \( f(1) \) is \( \log_{2} 1 = 0 \).
• Similarly, \( x = -1 \) has \( f(-1) = \log_{2} 1 = 0 \), utilizing symmetry.
• At \( x = 2 \) and \( -2 \), \( f(x) = \log_{2} 2 = 1 \).

Start by plotting the key points: \((1, 0), (-1, 0), (2, 1), (-2, 1)\).
Because the function is symmetric, mirror these points across the y-axis to complete the graph. Note that as \( x \) approaches zero from either direction, the graph plunges towards negative infinity, reflecting the logarithmic nature. Adding these aspects together creates a comprehensive depiction of \( \log_{2} |x| \) and illustrates graphing techniques effectively.