Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Graph each function. $$f(x)=\log _{2}\left(x^{2}\right)$$

Step-by-Step Solution

Answer

The graph of \( f(x) = \log_2(x^2) \) is an even function symmetric about the y-axis with key points (1,0), (2,2), (-2,2), decreasing to negative infinity as \( x \to 0 \), and increasing as \( |x| \to +\infty \).

Step 1: Understand the Domain

The domain of the function is determined by the expression inside the logarithm. Since we have a log function \[ f(x) = \log_2(x^2) \] we need \( x^2 > 0 \). This condition is met for all \( x eq 0 \). Thus, the domain of the function is all real numbers except zero.

Step 2: Identify Symmetry

Recall properties of even and odd functions. Since \( f(x) = \log_2(x^2) \), and \( x^2 \) is even, the function \( f(x) \) is also an even function. This means the graph will be symmetric about the y-axis.

Step 3: Calculate Key Points

Choose key points to plot the graph. For \( x = 1 \), \( f(1) = \log_2(1^2) = \log_2(1) = 0 \). For \( x = 2 \), \( f(2) = \log_2(2^2) = \log_2(4) = 2 \). For \( x = -2 \), \( f(-2) = \log_2((-2)^2) = \log_2(4) = 2 \). Note that \( f(x) = f(-x) \).

Step 4: Analyze Behavior

Since \( \log_2 \) function behavior is based on increasing or decreasing values of \( x^2 \), note that as \( x \to 0^+ \) or \( x \to 0^- \), \( f(x) \to -\infty \), and as \( |x| \to +\infty, f(x) \to +\infty \).

Step 5: Sketch the Graph

Using the key points and the analysis, plot the points (1,0), (2,2), and (-2,2). Draw a smooth curve that passes through these points while approaching near the x-axis as you move towards zero, and increasing rapidly as \( |x| \to +\infty \), keeping the symmetry around the y-axis.

Key Concepts

Graphing Logarithmic Functions

When it comes to graphing logarithmic functions, particularly those of the form such as \( f(x) = \log_2(x^2) \), it's important to start by identifying key characteristics. A logarithmic function transforms multiplication into addition, which gives its graph a distinctive, gentle slope. For this particular function, you would start by choosing a few key points. These could be \((1,0)\), \((2,2)\), and \((-2,2)\). These points help offer an understanding of how the curve behaves.

• Logarithmic graphs never cross the x-axis.
• They are undefined for negative values if within the basic logarithmic form, but since we are dealing with \(x^2\), the graph is defined for all non-zero numbers.
• The general shape resembles a stretched out 'S' starting from low values for small \(|x|\), climbing slowly and increasing as \(|x|\) grows.

Based on these points, you can sketch a graph that begins below the x-axis, touches at \( (1,0) \), and symmetrically mirrors this behavior in the other direction. The graph approaches infinity as \(|x|\) increases, displaying its characteristic logarithmic rise.

Domain and Range

Understanding the domain and range of logarithmic functions is crucial. For our function \( f(x) = \log_2(x^2) \), the domain comprises all values of \( x \) which make \( x^2 \) positive. Therefore, the function is defined for all real numbers except zero.

• Domain: \( \{ x \in \mathbb{R} \mid x eq 0 \} \)
• Range: All real numbers because for both extensions to negative and positive infinities, \( f(x) \) covers every real output value.

This domain and range ensure that while the function is undefined at zero (creating a point where the graph cannot be plotted), it can take any other real value both on the positive and negative axes, stretching infinitely in either direction.

Symmetry in Functions

Symmetry can be a helpful feature when analyzing functions. For the function \( f(x) = \log_2(x^2) \), symmetry manifests in its even nature. "Even" functions are those satisfying the condition \( f(x) = f(-x) \), meaning the graph is symmetric with respect to the y-axis.

• This symmetry simplifies graphing considerably, as it means only values on one side of the y-axis need to be calculated – you can mirror these for the other side.
• In our example, the points \( (2,2) \) and \( (-2,2) \) show how function values repeat on either side of the y-axis.
• Therefore, understanding this symmetry helps not just in graphing but in conceptual operations and calculations involving the function.

This feature is important as it can provide insight into how related phenomena could behave symmetrically in real-world applications.

Behavior of Functions

The behavior of the logarithmic function \( f(x) = \log_2(x^2) \) shifts based on positive or negative values moving towards infinity or closer to zero.

• As \( x \to 0^+ \) or \( x \to 0^- \), \( f(x) \to -\infty \), indicating a decline towards negative infinitum as we approach zero – this aligns with the undefined domain at zero and creates a classic logarithmic curve close to the y-axis.
• Conversely, as \( |x| \to +\infty \), \( f(x) \to +\infty \). This describes the increase of the graph as \(|x|\) grows.

These behaviors show the logarithmic characteristic of slow initial increase, rapidly growing after solid foundation around the zeroed point, and also predict the graph's extremities – where it starts and ends as \(|x|\) increases. All in all, comprehending function behaviors aids in anticipating future values and trends affected by the logarithmic influence.