Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The graph of \(f(x)=\log _{2} x\) approaches, but never touches, the negative portion of the \(y\) -axis. Thus the \(y\) - axis is an ____ of the graph.

Step-by-Step Solution

Answer

The \( y \)-axis is a vertical asymptote of the graph.

Step 1: Understanding the Behavior of Logarithmic Functions

The function given is \( f(x) = \log_{2} x \). Logarithmic functions have a domain of positive real numbers, \( x > 0 \). This means the function is only defined for positive values of \( x \) and approaches zero as \( x \) becomes very small from the right.

Step 2: Identifying the Asymptote

As \( x \to 0^{+} \), \( \log_{2} x \to -\infty \). This behavior indicates that the graph of the function gets closer and closer to the \( y \)-axis but never crosses or touches it.

Step 3: Determining the Type of Asymptote

Since \( f(x) \) approaches the \( y \)-axis (at \( x = 0 \)) but never reaches it, the \( y \)-axis acts as a vertical line that the graph approaches asymptotically. Therefore, the \( y \)-axis is a vertical asymptote for the graph of \( f(x) = \log_{2} x \).

Key Concepts

Vertical Asymptote

In the context of a logarithmic function, the term "vertical asymptote" is highly significant. A vertical asymptote is a vertical line that a graph approaches but never actually touches or crosses. For the function \( f(x) = \log_{2} x \), the vertical asymptote is the \( y \)-axis. This happens because as \( x \) approaches zero from the right (positive side), the output of the logarithmic function decreases towards negative infinity.

In simpler terms, no matter how close \( x \) gets to zero, \( f(x) \) will always continue to drop off steeply in value downwards. The concept of the vertical asymptote is foundational: it visualizes a boundary or limit for the function's behavior without the function ever crossing it.

Key takeaway: vertical asymptotes highlight where the function is undefined and often represent boundary behavior for many types of functions, especially rational and logarithmic functions.

Domain of Function

The domain of a function defines the set of all possible input values (\( x \)-values) for which the function is defined. For the logarithmic function \( f(x) = \log_{2} x \), the domain is \( x > 0 \). This means that only positive real numbers are valid inputs.

Here’s why: the logarithm is undefined for non-positive numbers such as zero or negative numbers because you cannot take the logarithm of zero or a negative number without moving into the realms of complex numbers.

• The domain excludes zero, at which the logarithmic value tends toward negative infinity.
• All positive real numbers are included, allowing for an expansive range of input values.

Understanding the domain is crucial because it helps us know what values \( x \) can assume for the function to return real-number outputs.

Behavior of Logarithmic Functions

Logarithmic functions like \( f(x) = \log_{2} x \) exhibit unique behaviors compared to linear and polynomial functions. One of the hallmarks of these functions is their slow rate of increase: as \( x \) becomes very large, \( f(x) \) rises, but very gradually.

Conversely, one of their distinguishing characteristics is their rapid descent as \( x \) approaches zero from the right. The function decreases steeply towards negative infinity, illustrating the vertical asymptote along the \( y \)-axis.

Key behavioral traits to note include:

• Logarithmic growth: The graph rises for increasing \( x \), approaching infinity, though slower than linear growth.
• Asymptotic fall: The function trends towards negative infinity as \( x \) nears zero on the positive side, adhering to its domain limitations.

By appreciating these behaviors, learners can better understand the dynamics and limitations of logarithmic functions.