Problem 134 - Elementary and Intermediate Algebra

Question

Graph: \(\quad f(x)=\log _{2}|x|\)

Step-by-Step Solution

Answer

The graph of \( f(x) = \log_{2} |x| \) has a vertical asymptote at \( x = 0 \) and is defined for all other real numbers.

Step 1: Understanding the Function

The function is defined as \( f(x) = \log_{2} |x| \). This signifies that you need to take the logarithm base 2 of the absolute value of \( x \). The function is defined for all \( x eq 0 \) because the logarithm of zero is undefined.

Step 2: Analyzing the Domain

The domain of \( f(x) = \log_{2} |x| \) is all real numbers except zero, \( x \in (-\infty, 0) \cup (0, \infty) \). This is because \(|x|\) must be positive to compute the logarithm.

Step 3: Determining the Range

The range of \( f(x) \) includes all real numbers. As \( |x| \) varies from a very small positive number to a large number, \( \log_2 |x| \) will take all possible real values from negative to positive infinity.

Step 4: Plotting the Graph

Sketch the graph by considering two cases: \( x > 0 \) and \( x < 0 \). For both, the graph is \( y = \log_{2} |x| \) shifted along the x-axis with \( (-\infty < x < 0) \) appearing as the reflection of \( (0 < x < \infty) \) due to the absolute value. The graph approaches negative infinity as \( x \to 0^{+} \) or \( x \to 0^{-} \), and it increases as \( |x| \to \infty \).

Step 5: Identifying the Asymptote

The vertical asymptote is at \( x = 0 \) since the function is undefined at 0. As \( x \) approaches 0 from either side, \( f(x) \to -\infty \).

Key Concepts

Function Domain

The domain of a function is the complete set of permissible input values (x-values) for which the function is defined. In the case of the logarithmic function given here, which is \( f(x) = \log_{2} |x| \), the domain is all real numbers except zero. This is due to the fact that the logarithm of zero is undefined, since you can't find a power to which 2 must be raised to get 0.

When dealing with absolute values, \(|x|\) represents the distance of \(x\) from zero on the number line, meaning it can never be negative. However, to use a logarithmic function, \(|x|\) must also be positive. If \(x = 0\), \(|x| = 0\) and it isn't defined in the domain of the logarithm.

So, the domain can be expressed as the union of two intervals:

• \(x \in (-\infty, 0)\)
• \(x \in (0, \infty)\)

For logarithmic functions, always ensure \(x eq 0\).

Range of Functions

The range of a function is the complete set of possible output values (y-values) that a function can produce. For the function \( f(x) = \log_{2} |x| \), the range is all real numbers. This is because the logarithm function, understanding the base 2 here, allows \( f(x) \) to stretch infinitely in both negative and positive directions depending on the value of \( x \).

When \(|x|\) is close to zero (but positive), the logarithmic value is negative and approaches negative infinity as \(x\) moves towards zero from either side. Conversely, as \(|x|\) becomes larger, \( f(x) \) becomes positive and stretches out towards positive infinity.

So, the range of the function can be expressed as:

• \(y \in (-\infty, \infty)\)

This wide range is typical of logarithmic functions due to their expansive nature in expressing continuous values.

Asymptotes of Graphs

Asymptotes are like invisible lines that a graph approaches but never touches or crosses. In the context of the logarithmic function \( f(x) = \log_{2} |x| \), the important asymptote is vertical, located at \( x = 0 \).

This vertical asymptote exists because the function is undefined at this point. As \(x\) approaches zero from either the positive or negative direction, \( f(x) \) dives down towards negative infinity. This means the graph is getting closer and closer to the y-axis, but it infinitely approaches without touching or crossing it.

A key point to remember is:

• The vertical asymptote at \(x = 0\) is due to the undefined nature of the logarithm at zero.

Understanding asymptotes helps explain why the graph behaves the way it does, giving you insights into the graph's general direction and limitations.

Graphing Logarithmic Functions

Graphing the function \( f(x) = \log_{2} |x| \) involves considering the behavior of the absolute value. Due to \(|x|\), the graph appears symmetric around the vertical line \(x = 0\), presenting a mirror image on both sides.

To effectively sketch the graph:

• Consider two primary regions: \(x > 0\) and \(x < 0\).
• Ensure to model \( y = \log_{2} |x| \) accurately for positive \( x \), which gradually increases, resembling typical logarithm behavior.
• Reflect this curve about the y-axis to depict the portion for negative \( x \), thanks to the absolute value.

The graph approaches negative infinity as \( x \to 0^{+} \) or \( x \to 0^{-} \) and increases continuously as \(|x|\) grows. Remember, the steep descent near zero is due to the logarithmic nature, and the reflection is a hallmark of incorporating absolute values.