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Chapter 4: Problem 19

Use a graphing utility to graph \(f\) \(g,\) and \(y=x\) in the same viewing window to verify geometrically that \(g\) is the inverse function of \(f .\) (Be sure to restrict the domain of \(f\) properly.) $$f(x)=\cos x, \quad g(x)=\arccos x$$

Short Answer

Expert verified
After graphing \(f(x) = \cos x\) (with restricted domain), \(g(x) = \arccos x\), and \(y = x\), you will see that \(f\) and \(g\) are mirror images of each about the line \(y = x\), thus indicating \(g(x) = \arccos x\) is the inverse of \(f(x) = \cos x\).

Step by step solution

01

Understand the inverse function property

An important property of inverse functions is that when graphed, if the function \(f\) is reflected across the line \(y = x\), it results in its inverse function \(g\). In other words, the graphs of the function \(f\) and its inverse function \(g\) are mirror images of each other with respect to the line \(y = x\).
02

Restrict the domain of \(f\)

Since \(\cos x\) oscillates from \(-\infty\) to \(+\infty\), there would be multiple \(x\) values for a single \(y\) value, which contradicts the function property. Therefore, to make it a function for the purpose of graphing, the domain must be restricted. Choose the interval [0, 蟺], which is the standard restricted domain for the \(\cos\) function to make it invertible.
03

Graph \(f(x) = \cos x\), \(g(x) = \arccos x\), and \(y = x\)

Use a graphing utility to graph these three functions in the same viewing window. \(f\) and \(g\) should reflect each other across the line \(y = x\), showing that these two functions are indeed inverses of each other.

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Most popular questions from this chapter

A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by \(y=\frac{1}{4} \cos 16 t, t>0,\) where \(y\) is measured in feet and \(t\) is the time in seconds. (a) Graph the function. (b) What is the period of the oscillations? (c) Determine the first time the weight passes the point of equilibrium \((y=0)\)

A carousel with a 50 -foot diameter makes 4 revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed (in feet per minute) of the platform rim of the carousel.

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\csc x$$

A ship leaves port at noon and has a bearing of \(\mathrm{S} 29^{\circ} \mathrm{W}\). The ship sails at 20 knots. (a) How many nautical miles south and how many nautical miles west will the ship have traveled by 6: 00 P.M.? (b) At 6: 00 e.m., the ship changes course to due west. Find the ship's bearing and distance from the port of departure at 7: 00 P.M.

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{4} \sin 6 \pi t$$
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