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Chapter 4: Problem 90

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

Short Answer

Expert verified
To sketch the function \(f(x)=\arccos \frac{x}{4}\), firstly understand the fundamental character of \(\arccos\). Then, determine the domain to be \([-4,4]\) and range to be \([0,\pi]\). Plot significant points within the domain like (4, 0), (0, \(\frac{\pi}{2}\)), and (-4, \(\pi\)); and draw a smooth curve through them.

Step by step solution

01

Understanding the function

Firstly, we should understand what \(\arccos(x)\) means. This is the inverse cosine function. It reverses the operation of the cosine function. For a given value in its domain, it gives the angle whose cosine is that value. The typical domain for \(y=\arccos(x)\) is \([-1,1]\), and the range is \([0, \pi]\). The graph of \(y=\arccos(x)\) decreases from \((1,0)\) to \((-1,\pi)\) which is different from the cosine function that oscillates between -1 and 1. Now, our function is \(f(x) = \arccos(\frac{x}{4})\). That division by 4 will stretch the graph along the x-axis.
02

Determine the domain and range of the given function

Next, identify the domain and range of the given function. The domain of \(f(x)=\arccos \frac{x}{4}\) is the set of all \(x\) such that \(-1 \leq \frac{x}{4} \leq 1\). Solving this inequality gives us a domain of \(-4 \leq x \leq 4\). Due to the \(\arccos\) function, the range remains unchanged. Thus, the range is \([0, \pi]\).
03

Plotting the graph

Finally, plot the graph. To accurately plot, take a few significant points within the domain, like -4, 0, and 4 - and calculate corresponding \(y\) values. For -4, \(y=\arccos(\frac{-4}{4})=\pi\). For 0, \(y=\arccos(0)=\frac{\pi}{2}\). And for 4, \(y=\arccos(\frac{4}{4})=0\). Plot these points and sketch a curve through them. The graph starts from (4, 0), increases until it reaches (0, \(\frac{\pi}{2}\)), and then decreases until it reaches (-4, \(\pi\)). It should lay within the region defined by the domain and range.

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Key Concepts

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

Graphing Functions
Graphing functions can be a fun yet puzzling task. It's like painting a picture of how the function behaves across different values. Let's dive into graphing the inverse trigonometric function:
  • Start by identifying key points. For \( f(x) = \arccos \frac{x}{4} \), the key points derived are: (-4, \( \pi \)), (0, \( \frac{\pi}{2} \)), and (4, 0).
  • Notice how the curve changes between these points. For example, it starts at the highest point (4, 0) and moves down to (-4, \( \pi \)).
  • By connecting these points smoothly, you can see how the graph is drawn, keeping the trajectory inside the calculated range.
Each of these points helps us understand the path the graph will take, much like waypoints on a map. This ensures that the shape we draw mirrors the true nature of the function.
Domain and Range
Understanding the domain and range of a function is critical when dealing with graphing and mathematical analysis. It essentially tells us what values are allowed to go into (domain) and come out of (range) a function.
  • The domain of \( f(x) = \arccos \frac{x}{4} \) is \([-4, 4]\). This arises from solving \(-1 \leq \frac{x}{4} \leq 1\), which gives us all possible \(x\) values that won't make the \(\arccos\) function undefined.
  • The range refers to potential \(y\) values the function can output, which remains the same as the standard \(\arccos\) range, \([0, \pi]\), regardless of the transformation applied to \(x/4\).
Both domain and range are central in graphing functions as they set the boundaries for feasible graphing and mapping.
Arccos Function
The \(\arccos\) function is one of the fundamental inverse trigonometric functions. It's the flip side of the \(\cos\) function, unraveling an angle from a cosine value.
  • The \(\arccos(x)\) gives us an angle where cosine results in \(x\). Think of it as reversing the cosine process.
  • Unlike the periodic nature of \(\cos(x)\), \(\arccos(x)\) is a one-to-one function with a specific range of \([0, \pi]\).
  • By applying it in \( f(x) = \arccos \left(\frac{x}{4}\right)\), the function stretches and scales along the x-axis, but fundamentally, it still operates by finding that unique angle with the given cosine value.
This identity of inverting cosine offers valuable insights and utility in various mathematical and real-world applications.

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Most popular questions from this chapter

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to determine whether the expressions are equivalent. Verify the results algebraically. $$ y_{1}=\sec ^{2} x-1, \quad y_{2}=\tan ^{2} x $$

PATTERN RECOGNITION (a) Use a graphing utility to graph each function. $$ \begin{array}{l} y_{1}=\frac{4}{\pi}\left(\sin \pi x+\frac{1}{3} \sin 3 \pi x\right) \\ y_{2}=\frac{4}{\pi}\left(\sin \pi x+\frac{1}{3} \sin 3 \pi x+\frac{1}{5} \sin 5 \pi x\right) \end{array} $$ (b) Identify the pattern started in part (a) and find a function \(y_{3}\) that continues the pattern one more term. Use a graphing utility to graph \(y_{3}\) (c) The graphs in parts (a) and (b) approximate the periodic function in the figure. Find a function \(y_{4}\) that is a better approximation.

Using calculus, it can be shown that the secant function can be approximated by the polynomial $$\sec x \approx 1+\frac{x^{2}}{2 !}+\frac{5 x^{4}}{4 !}$$ where \(x\) is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare?

Sketch the graph of the function. Include two full periods. $$ y=-2 \sec 4 x+2 $$

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arcsin (-0.75) $$
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