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Chapter 1: Problem 33

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

Short Answer

Expert verified
The graph of the function \(f(x)= \ln(2x)\) is a compression of the graph of the parent function \(f(x)= \ln(x)\) towards the y-axis by a factor of 2. The domain is \(x > 0\).

Step by step solution

01

Identify the Parent Function and Translation

The parent function here is \(f(x) = \ln x\), and it is being horizontally translated by a factor of 2. That means, the function \(f(x) = \ln(2x)\) will be compressed horizontally compared to the parent function.
02

Plot the Parent Function

Start by plotting the parent function \(f(x) = \ln x\) which has a vertical asymptote at \(x = 0\), passes through the point (1,0), and grows slowly for \(x > 1\). The graph extends infinitely on the y-axis and its direction is always positive on the x-axis.
03

Construct the Translated Function

To plot \(f(x) = \ln(2x)\), compress the x-values by half. This means every point \((x,y)\) on \(f(x) = \ln x\) transforms to \((x/2, y)\) on \(f(x) = \ln(2x)\). The result is a compression of the original graph towards the y-axis.
04

State the Domain

The domain of the function, like all logarithmic functions, is defined for strictly positive values. Therefore, for \(f(x) = \ln(2x)\), the domain is \(x > 0\).

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