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Chapter 4: Problem 89

Use a graphing utility to graph the function. $$f(x)=2 \arccos (2 x)$$

Short Answer

Expert verified
The graph of the function \(f(x) = 2 \arccos (2x)\) is a decreasing curve with its domain from -1 to 1 and its range is from 0 to \(2\pi\).

Step by step solution

01

Identify the function

Given is the function \(f(x) = 2 \arccos(2x)\). The function combines two transformations: multiplication by 2 which scales the output the arccos function, and the transformation of the input of the arccos function.
02

Use a Graphing tool

Input the function \(f(x) = 2 \arccos (2x)\) into a graphing calculator or tool. Note that the domain of the inverse cosine function is from -1 to 1, thus the same applies to the input of our function 2x reaching from -1 to 1.
03

Analyze the graph

Observe the results from the graphing tool. The function \(f(x) = 2 \arccos (2x)\), shares the similar pattern with \(\arccos x\), but it ranges from \(0\) to \(2\pi\) rather than from \(0\) to \(\pi\), since there is a scaling factor of 2 applied to \(\arccos(2x)\). This effect is due to the 2x within the arccos, which causes a horizontal compression of the graph by a factor of 2. The graph should show a decreasing curve between -1 and 1, showing that \(f(x)\) decreases as \(x\) increases.

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