Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$ f(x)=\log _{3}\left(x^{2}\right) $$

Step-by-Step Solution

Answer

The graph of \(f(x) = \log_{3}(x^2)\) is symmetric about the y-axis and resembles a 'V' shape opening upwards.

Step 1: Understand the Function Form

The given function is \(f(x) = \log_{3}(x^2)\). This is a logarithmic function with base 3. It represents the logarithm of \(x^2\) with respect to base 3. The function is defined for \(x eq 0\) (as \(x^2 > 0\) for all real numbers except zero).

Step 2: Analyze Domain and Range

The domain of \(f(x) = \log_{3}(x^2)\) is all real numbers except 0, which means \((-\infty, 0) \cup (0, \infty)\). The range of a logarithmic function is all real numbers \((-\infty, \infty)\) because \(x^2\) can take any positive value.

Step 3: Calculate Key Points

Let's calculate some key points for plotting. Since \(f(x) = \log_3((x^2)) = 2 \cdot \log_3(|x|)\), when \(x = \pm 1\), \(f(x) = \log_3(1) = 0\). Similarly, when \(x = \pm 3\), \(f(x) = \log_3(9) = 2\).

Step 4: Determine Function Symmetry

The function \(f(x) = \log_{3}(x^2)\) is even because \(f(-x) = \log_{3}((-x)^2) = \log_{3}(x^2) = f(x)\). This shows it has symmetry about the y-axis.

Step 5: Sketch the Graph

On the graph, plot the points \((-3, 2), (-1, 0), (1, 0), (3, 2)\). The graph approaches \(-\infty\) as \(x\) approaches 0 from either side due to the logarithmic function behavior. The graph is symmetric about the y-axis and continually rises as \(|x|\) increases. This gives a 'V'-shaped plot opening upwards since \(f(x)\) is defined for both positive and negative \(x\) excluding zero.

Key Concepts

Domain and Range

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 in Functions

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.

Graphing Logarithmic Functions

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.