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Chapter 4: Problem 88

Sketch a graph of the function. $$f(x)=\arccos \frac{x}{4}$$

Short Answer

Expert verified
The graph of function \(f(x)=\arccos \frac{x}{4}\) decreases from \(\pi\), when \(x=-4\), to 0, when \(x=4\). The domain of function is [-4, 4] and the range is [0, \(\pi\)].

Step by step solution

01

Understand the Key Properties of the Arccos function

The range of the \(\arccos(x)\) function is [0, \(\pi\)]. The domain is [-1,1]. Therefore, for the function \(f(x)=\arccos \frac{x}{4}\), only values of \(x\) which make \(\frac{x}{4}\) in the interval [-1,1] are in the domain. That is \(x\) values from [-4,4]. Also, since the principal range of the arccosine function is \([0, \pi]\), \(f(x)\) values lie in the interval [0, \(\pi\)].
02

Understand Denominator Impact

\(\frac{x}{4}\) expands the original range of -1 to 1 of the arccos function, to be between -4 and 4. Beyond these values, the function is undefined. The effect of the denominator under the arccos function is to stretch the graph horizontally by a factor of 4.
03

Plotting the Function

Plot the function knowing that the function is undefined for \(x<-4\) or \(x>4\), that y-values range from 0 to \(\pi\) and that the function has a horizontal stretch factor of 4. Start by plotting two key points: \((-4, \pi)\) and \((4, 0)\). Then, decreases as x increases.
04

Finalize the Graph

Complete the graph by making sure it is a smooth, continuous curve that passes through the two key points plotted. The graph should start at the point (-4, \(\pi\)) and end at the point (4, 0). Ensure the graph lies completely within the defined intervals

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Most popular questions from this chapter

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