Problem 64 - College Algebra with Modeling and Visualization

Question

Sketch a graph of \(f\) $$f(x)=\log _{2} x^{2}$$

Step-by-Step Solution

Answer

The graph of \( f(x) = \log_2(x^2) \) is symmetric about the y-axis, has vertical asymptotes at \( x = 0 \), and both ends approach infinity as \(|x|\) increases.

Step 1: Understand the Logarithmic Function

The given function is a logarithmic function, specifically an expression of the form \( f(x) = \log_2(x^2) \). This format implies that \( f(x) \) represents the logarithm to base 2 of \( x^2 \).

Step 2: Simplify the Function

Using the logarithmic identity \( \log_b(a^c) = c \cdot \log_b(a) \), simplify \( f(x) \) as follows: \[ f(x) = \log_2(x^2) = 2 \cdot \log_2(x) \]. This can help in understanding that the graph of \( f(x) \) will behave like twice the standard \( \log_2(x) \) function.

Step 3: Identify the Domain

For the function \( f(x) = \log_2(x^2) \), \( x^2 \) must be positive, which is always true except at \( x = 0 \). Thus, the domain of \( f(x) \) is all real numbers except \( x = 0 \).

Step 4: Determine Key Features

Determine key points and behavior of the function. Notably:- At \( x=1 \), \( f(1) = \log_2(1^2) = \log_2(1) = 0 \).- At \( x=-1 \), \( f(-1) = \log_2((-1)^2) = \log_2(1) = 0 \).- As \( x \to 0^+ \), \( f(x) \to -\infty \).- As \( x \to 0^- \), \( f(x) \to -\infty \).- For large \( |x| \), \( f(x) \) tends to \( \infty \) as both positive and negative values approach infinity.

Step 5: Sketch the Graph

Plot the key points and behaviors on a coordinate plane:- Plot the point (1, 0) and (-1, 0).- As \( x \) approaches 0 from either direction, the graph goes down to negative infinity, indicating vertical asymptotes on both sides approaching \( x = 0 \).- For \( x > 0 \), the graph looks like the log base 2 graph but rising more steeply.- Reflect that shape across the y-axis due to the even power, resulting in symmetry along the y-axis.

Key Concepts

Domain of Functions

Understanding the domain of a function is crucial. For the function \(f(x) = \log_{2}(x^2)\), the domain refers to the set of all possible \(x\) values the function can accept. In a logarithmic function, we can only log positive numbers. Here, because of the \(x^2\) term, \(x^2\) is always positive except when \(x = 0\).

Thus, the domain of the function is all real numbers except zero. In interval notation, this is written as \((-\infty, 0) \cup (0, \infty)\). Being aware of the domain helps avoid mathematical errors when evaluating or graphing functions.

Graphing Techniques

Graphing logarithmic functions involves several techniques, especially when transformations are involved. For \(f(x) = \log_{2}(x^2)\), we first simplify it using logarithmic identities. This gives \(f(x) = 2 \cdot \log_{2}(x)\), an amplified version of the basic \(\log_{2}(x)\) graph, meaning it stretches vertically.

Key graphing techniques include:

• Identifying intercepts: Notice that \(f(1) = 0\) and \(f(-1) = 0\), meaning the graph crosses the x-axis at these points.
• Noting asymptotic behavior: As \(x\) approaches zero from either side, the graph goes down towards \(-\infty\), indicating vertical asymptotes at \(x = 0\).
• Extending behavior: For \(x\) far from zero, the graph continues to rise, reflecting the function's tendency towards infinity as \(x\) becomes very large or very small in magnitude.

Graphing these features provides a clear visual representation of the function's behavior.

Logarithmic Identities

Logarithmic identities are tools that simplify expressions and calculations. For \(f(x) = \log_{2}(x^2)\), using the identity \(\log_b(a^c) = c \cdot \log_b(a)\) results in \(f(x) = 2 \cdot \log_{2}(x)\).

This identity shows:

• How powers on a variable within a log function can become a coefficient, simplifying analysis.
• The importance of simplification for identifying the transformed shape of the standard logarithmic graph, enhancing graphing ease.

Understanding these identities expands your ability to manipulate and comprehend complex expressions, a key skill in advanced mathematics.

Symmetry in Functions

The concept of symmetry simplifies both the graphing and interpretation of functions. For \(f(x) = \log_{2}(x^2)\), note how squaring \(x\) influences symmetry.

A function is symmetric with respect to the y-axis if substituting \(-x\) into the function yields the original function, that is, \(f(-x) = f(x)\). Here, \(f(x) = \log_{2}(x^2)\) ensures that \(f(-x) = \log_{2}((-x)^2) = \log_{2}(x^2)\), showing y-axis symmetry.

This symmetry indicates that the graph mirrors itself on either side of the y-axis, aiding in sketching and understanding the graph's full shape with less effort by plotting half and reflecting. Recognizing such symmetries not only makes graphing easier but also reveals deeper insights into the function’s properties.