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Chapter 6: Problem 62

Use a graphing calculator to graph the function. Determine the domain, range, and asymptote of the function. \(y=-\ln x\)

Short Answer

Expert verified
The domain of the function \(y= -\ln x\) is \(x > 0\). The range is all real numbers{-\infty, +\infty}. The function has a vertical asymptote at \(x = 0\).

Step by step solution

01

Graph the Function

Using your graphing calculator, plot the function \(y = - \ln x\). You will observe that the graph is a reflection of the graph of \(y = \ln x\) over the x-axis.
02

Identify the Domain

The domain of a function is the set of all possible x-values. Because the logarithmic function is undefined for values less than or equal to zero, the domain of this function is \(x > 0\).
03

Identify the Range

The range of a function is the set of all possible y-values. Looking at the graph, it can be seen that the y-values can take on any number, so the range of this function is all real numbers{-\infty, +\infty}.
04

Determine the Asymptote

An asymptote is a line the graph gets arbitrarily close to, but never actually reaches. For this function, as x approaches 0 from the right, \(y = - \ln x\) approaches negative infinity, thus the vertical line \(x = 0\) is a vertical asymptote.

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Key Concepts

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

Domain of a Function
The domain of a function is a critical concept in mathematics that describes all possible input values, or x-values, that will produce a valid output. For logarithmic functions like \(y = -\ln x\), this characteristic is particularly significant as it impacts where the function is defined.
  • Logarithms are undefined for zero and negative numbers because you cannot take the logarithm of such values in the real number system.
  • Therefore, the domain of \(y = -\ln x\) must be limited to positive values.
  • This means the domain is \(x > 0\).
This domain restriction is essential to remember when working with logarithmic functions, as it dictates the portions of the graph we can calculate and visualize.
Range of a Function
The range of a function refers to all possible output values, or y-values, a function can produce. For the graph of \(y = -\ln x\), this concept becomes clearer by closely examining the behavior of the graph.
  • Since the logarithmic function \(\ln x\) produces numbers ranging over all real numbers as x moves from 0 to infinity, its reflection, \(-\ln x\), would have the same range but negative values.
  • Therefore, the range of \(y = -\ln x\) is \((-\infty, \infty)\).
The infinite nature of the range implies that as x increases, the values produced can extend indefinitely in the negative direction, while still covering an extensive span across all negative real numbers.
Asymptotes
Asymptotes are essential lines that graphs of certain functions approach but never touch. They often present themselves in logarithmic functions and provide important insights into the behavior of these functions.
  • For the function \(y = -\ln x\), the vertical asymptote is a key feature.
  • The graph nears the vertical line at \(x = 0\) without ever actually meeting it.
  • As x approaches zero from the right, the y-values plummet towards negative infinity, highlighting the presence of this vertical asymptote.
Understanding the asymptotic behavior can aid in predicting how the graph behaves at extreme values and sharpen our comprehension of the function's overall structure.

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