91Ó°ÊÓ

When dealing with logarithmic functions, understanding domain and range is crucial. For the function \( f(x) = \log_{3}(x^2) \), the expression inside the logarithm, \( x^2 \), necessitates that \( x eq 0 \) since we cannot take the logarithm of zero. This results in a domain of all real numbers except zero, written mathematically as \((-fty, 0) \cup (0, \infty)\).The range of a function describes all possible values that the function can output. For logarithmic functions like \( f(x) \), the range is all real numbers \((-fty, \infty)\). Even though \( x^2 \) takes only positive values, \( \log_{3} \) of these values can cover the entire spectrum of real numbers. Thus, the range remains unaffected, spreading from negative to positive infinity.

Symmetry is a beautiful property in functions that can make understanding and graphing easier. A function is **even** if it is symmetric about the y-axis. This occurs when \( f(x) = f(-x) \) for all \( x \) in the domain. For the function \( f(x) = \log_{3}(x^2) \), substituting \( -x \) for \( x \) results in \( f(-x) = \log_{3}((-x)^2) = \log_{3}(x^2) = f(x) \). This equation confirms that the function is even, indicating symmetry about the y-axis. Thus, for every point \( (x, y) \) on the graph, there is a corresponding point \( (-x, y) \) reflecting across the vertical line at the origin. This property helps when plotting or analyzing the function visually, as it reduces the need to calculate values for negative \( x \) separately.

Sketching the graph of a logarithmic function involves understanding its basic shape and significant characteristics. For \( f(x) = \log_{3}(x^2) \), - Start by identifying key points. The points \( (-1, 0), (1, 0), (-3, 2), (3, 2) \) are crucial as they provide a framework for the graph.- As \( x \) approaches 0 from either direction, \( f(x) \) trends towards \(-\infty\). This behavior is typical for logarithmic functions, representing a vertical asymptote at \( x = 0 \).- The function is even, meaning it's symmetric about the y-axis; you can mirror any part of the graph across this axis for a complete picture.With these elements, plot the points on a graph, noting the symmetry and the downward trend as \( x \) nears zero. The graph resembles a 'V' shape, opening upwards. The function rises steeply as \(|x|\) increases on either side, perfectly showcasing the distinctive nature of logarithmic functions.