/*! 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 85 Sketch a graph of the function. ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 85

Sketch a graph of the function. $$f(x)=\arctan 2 x$$

Short Answer

Expert verified
The graph of \(f(x)=\arctan 2x\) is similar to the standard arctan function, but it is horizontally compressed by a factor of 0.5. It has horizontal asymptotes at \(y = \pi/2\) and \(y = -\pi/2\), and is symmetric with respect to the origin.

Step by step solution

01

Understand the Basic Properties of the arctan Function

The arctan function has following properties: \n1. Domain: All real numbers.\n2. Range: \(-\pi/2\) to \(\pi/2\).\n3. The function is continuous everywhere.\n4. Two horizontal asymptotes, \(y = \pi/2\) and \(y = -\pi/2\).\n5. It is an odd function, meaning it is symmetric over the origin.
02

Apply the Basic Transformations

The function to be graphed is \(f(x) = \arctan 2x\). The 2x inside the function creates a horizontal stretch by a factor of 0.5 or equivalently a frequency change. As frequency is the reciprocal of period, the function \(f(x) = \arctan 2x\) experiences a frequency doubling compared to \(f(x) = \arctan x\). This means it takes less time (less x) for \(f(x) = \arctan 2x\) to complete one cycle from \(-\pi/2\) to \(\pi/2\).
03

Sketch the Graph

To sketch a graph: \n1. Start by drawing the horizontal asymptotes at \(y = \pi/2\) and \(y = -\pi/2\).\n2. Sketch the curve slightly stretched horizontally. It should have a higher frequency. \n3. The graph should be symmetric with respect to the origin.

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