/*! 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 134 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 134

Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arccos \frac{x}{4} $$

Short Answer

Expert verified
The graph of \(f(x) = \arccos \(\frac{x}{4}\)\) starts from point (-4, 蟺), reaches a peak at (0, 蟺/2), and descends to the point (4, 0). This makes the plot decrease across its domain from -4 to 4.

Step by step solution

01

Determine the Domain and Range

First, the domain and range of the given function can be determined. For the arccosine function \(f(x) = \arccos x\), the domain is restricted between -1 and 1 inclusive (i.e. \(x \in \[-1, 1\]\)), and the range falls between 0 and 蟺 inclusive (i.e. \(y \in \[0, 蟺\]\)). But for the function \(f(x) = \arccos \(\frac{x}{4}\)\), the domain will fall between -4 and 4 inclusive because the input is now \(x/4\). The range remains the same as the original arccosine function and falls between 0 and 蟺 inclusive.
02

Plot Key Points

Briefly plot key points. Start from the endpoints of the domain, which are -4 and 4. When \(x = -4\), \(f(x) = \arccos(-1) = 蟺\). When \(x = 4\), \(f(x) = \arccos(1) = 0\). Another key point to note is when x = 0, then \(f(x) = \arccos(0) = 蟺/2\). So plot these three points: (-4, 蟺), (0, 蟺/2), and (4, 0).
03

Sketch the Graph

Finally, sketch the graph of the function. The graph starts from the point (-4, 蟺), reaches a peak at (0, 蟺/2) and then descends to the point (4, 0). It's important to note that the graph of the arccosine function is decreasing on its domain.
04

Verification with a Graphing Utility

Now input the function \(f(x) = \arccos \(\frac{x}{4}\)\) into a graphing utility to verify whether the sketch matches the curve produced by the graphing utility.

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