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Chapter 1: Problem 32

Sketch the graph of the function and state its domain. $$ f(x)=-2 \ln x $$

Short Answer

Expert verified
The graph is a downward-opening curve starting at (1,0) and extending towards \(\infty\) on the x-axis from the point (e,-2). The domain of the function is \( (0, \infty) \).

Step by step solution

01

Identify the Parent Function

The function \( f(x) = -2\ln x \) is a transformation of the basic function \( f(x) = \ln x \).
02

Determine the Transformation

The coefficient -2 in front of the logarithm implies two transformations: 1) Reflection in the x-axis and 2) Vertical stretch by a factor of 2.
03

Plot the Graph using the Transformation

Before applying the transformation, the graph of \( \ln x \) includes the points (1,0) and (e,1). After applying the transformation, these points become (1,0) and (e,-2). The curve of the graph will extend from these points towards \(\infty\) on the x-axis but never touch the x-axis, reflecting over the x-axis due to the negative coefficient, creating a downward-opening curve.
04

Define the Domain

Logarithmic functions are undefined for negative values and zero. So, the domain of the function is \( (0,\infty) \).

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