/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 Sketch the graph of \(f\). $$ ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 45

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

Short Answer

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

Step by step solution

01

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.
02

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: \( x > 0 \).
03

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 \( x = 0 \).
04

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 \( \log_{3}\left(\frac{1}{x}\right) < 0 \).
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檛 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.

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