Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$ f(x)=\log _{3}\left(\frac{1}{x}\right) $$

Step-by-Step Solution

Answer

The graph has a vertical asymptote at \( x=0 \) and crosses the x-axis at \( x=1 \).

Step 1: Understand the Function

The function given is \( f(x) = \log_{3}\left(\frac{1}{x}\right) \). This is a logarithmic function base 3, where the argument is reciprocal of \( x \). We're interested in how the reciprocal impacts the behavior of a typical logarithmic graph.

Step 2: Explore the Domain

Since \( \log_{3}\left(\frac{1}{x}\right) \) is defined only when \( \frac{1}{x} > 0 \), it implies that \( x \) must be greater than 0. Thus, the domain of \( f(x) \) is all positive real numbers: \( x > 0 \).

Step 3: Determine the Asymptotes

Logarithmic functions typically have vertical asymptotes where their arguments approach 0. Because \( \frac{1}{x} \to \infty \) as \( x \to 0^+ \), there is a vertical asymptote at \( x = 0 \).

Step 4: Analyze Intervals

When \( x = 1 \), \( \frac{1}{x} = 1 \), so \( f(1) = \log_{3}(1) = 0 \). For \( x < 1 \), \( \frac{1}{x} > 1 \), so \( \log_{3}\left(\frac{1}{x}\right) > 0 \). For \( x > 1 \), \( \frac{1}{x} < 1 \), so \( \log_{3}\left(\frac{1}{x}\right) < 0 \).

Step 5: Sketch the Graph

Plot the key points and asymptotes: the point \((1, 0)\) lies on the graph, and there is a vertical asymptote at \( x = 0 \). Since \( f(x) > 0 \) for \( x < 1 \) and \( f(x) < 0 \) for \( x > 1 \), the graph starts above the x-axis to the left of \( x = 1 \), crosses at \( x = 1 \), and decreases further below the x-axis for \( x > 1 \). The graph approaches the vertical asymptote but never crosses it.

Key Concepts

Asymptotes

An important feature of the logarithmic function \( f(x) = \log_{3}\left(\frac{1}{x}\right) \) is its asymptotic behavior. In mathematics, asymptotes help us understand how functions behave as inputs get really large or really small. For the function given, the vertical asymptote is crucial.

• Vertical asymptotes occur where the function tends to infinity as it approaches a particular x-value.
• For \( f(x) = \log_{3}\left(\frac{1}{x}\right) \), the vertical asymptote is at \( x = 0 \).
• This is because as \( x \to 0^+ \), \( \frac{1}{x} \to \infty \).

The graph of \( f(x) \) will approach this line but never actually meet it. One can see the graph flying off towards infinity near \( x = 0 \), clearly illustrating the asymptote.

Function Domain

Understanding the domain is crucial for functions like \( f(x) = \log_{3}\left(\frac{1}{x}\right) \), which have specific requirements for their inputs to be valid.

• The logarithm function is only defined when its argument is greater than zero.
• Since the argument here is \( \frac{1}{x} \), the function is defined for \( \frac{1}{x} > 0 \).
• This requirement implies that \( x \) must be greater than zero. Therefore, the domain is \( x > 0 \).

This means the graph of \( f(x) \) will only exist in the positive x-region, and we won’t find any part of the curve in the negative x values.

Reciprocal Function Behavior

The reciprocal within the logarithmic function, \( \frac{1}{x} \), plays a significant role in dictating the behavior of the graph.

• The reciprocal \( \frac{1}{x} \) becomes larger as \( x \) approaches zero, and smaller as \( x \) increases.
• This characteristic results in inverted behavior compared to a simple logarithm \( \log_{3}(x) \).
• When \( x < 1 \), \( \frac{1}{x} > 1 \), making \( \log_{3}(\frac{1}{x}) > 0 \).
• When \( x > 1 \), \( \frac{1}{x} < 1 \), so \( \log_{3}(\frac{1}{x}) < 0 \).

This tells us that the graph, instead of increasing as a standard logarithm would, starts high and decreases past zero, following a path that reflects through the basic log function's behavior.