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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 sketch of the function \(f(x) = \arccos \frac{x}{4}\) is a horizontally stretched version of the inverse cosine function, ranging from -4 to 4 on the x-axis and from 0 to \(\pi\) on the y-axis.

Step by step solution

01

Identifying the Function

The function \(f(x)=\arccos \frac{x}{4}\) is an inverse cosine function that has been modified. So, the parent function is the inverse cosine function \(y=cos^{-1}(x)\). The modification is the division of x by 4 inside the function, which stretches the graph horizontally by a factor of 4.
02

Setting Up the Axes

The next step is to set up the x and y axes. Since the x inside the inverse cosine function is divided by 4, this implies x will range from -4 to 4 because cosine function is defined between -1 and 1, hence, \(-1 \leq \frac{x}{4} \leq 1\). Also, the arccosine function is defined on the interval \([0, \pi]\). Thus, the y-values will range from 0 to \(\pi\).
03

Plotting Points

Finally, some important points can be plotted. When x is 0, \(f(x)=\arccos 0 = \frac{\pi}{2}\). When x is -4, \(f(x)=\arccos (-1)= \pi \). When x is 4, \(f(x)=\arccos 1 = 0\). Connecting these points will result in the graph of the function \(f(x)=\arccos \frac{x}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Transformations
Function transformations involve modifying a basic function to produce a new function. When we look at the given function \(f(x) = \arccos \frac{x}{4}\), we can see a transformation has occurred. The parent function here is \(y = \arccos(x)\), a standard inverse cosine function.

To achieve the transformation, we divide the input \(x\) by 4. This division stretches the graph of the inverse cosine function horizontally. Here鈥檚 how it works:
  • With \(\arccos(x)\), the function accepts values between -1 and 1 for \(x\), but \(\frac{x}{4}\) alters this to accept values between -4 and 4.
  • This means the x-values spread out more, resulting in a horizontal stretch by a factor of 4.
Understanding transformations like these is key to correctly graphing altered or transformed functions.
Graphing Functions
Graphing the function \(f(x) = \arccos \frac{x}{4}\) involves understanding its behavior and plotting key points accurately. The inverse cosine function, by nature, produces angles whose cosine is the input value. In this case, the function inputs the value \(\frac{x}{4}\) rather than \(x\) directly.

Here's a step-by-step method to graph it:
  • Determine the critical points: At \(x = 0\), \(f(x) = \arccos 0 = \frac{\pi}{2}\). For \(x = -4\), \(f(x) = \arccos (-1) = \pi\). Lastly, at \(x = 4\), \(f(x) = \arccos 1 = 0\).
  • Connect these points: When plotted on the graph, these points outline the arccosine curve, stretched along the x-axis. The curve will gently move downwards from \((0, \frac{\pi}{2})\), reaching \((4, 0)\), and upwards to \((-4, \pi)\).
The graph should be smooth and reflect the stretched effect caused by dividing the inside value by 4.
Domain and Range
In any function, domain and range describe where the function's input and output values lie. For the function \(f(x) = \arccos \frac{x}{4}\), stating the domain and range helps us understand its limits on a graph.
  • Domain: Refers to all possible x-values that the function can accept. Since the formula \(\arccos(\frac{x}{4})\) requires \(-1 \leq \frac{x}{4} \leq 1\), this translates to the domain being \([-4, 4]\). This indicates that \(x\) values can vary between -4 and 4.
  • Range: Refers to all possible values the function can output. The arccosine function outputs angles in the range \([0, \pi]\). Consequently, \(f(x)\) will similarly have a range of \([0, \pi]\), meaning the function outputs angle measures from 0 to \(\pi\).
Knowing the domain and range is crucial for correctly setting up the axes and interpreting the function's behavior.

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