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Chapter 4: Problem 101

How can the graph of \(y=\sin ^{-1} x\) be obtained from the graph of the restricted sine function?

Short Answer

Expert verified
The graph of \(y=\sin^{-1} x\) can be obtained from the graph of the restricted sine function by reflecting it over the line \(y=x\). The resulting graph rises from \(-\pi/2\) to \(\pi/2\) as x runs from -1 to 1.

Step by step solution

01

Understanding the Functions

The sine function, \(y = \sin x\), is periodic and not one-to-one, meaning it doesn't pass the horizontal line test. Therefore, to create an inverse function, it has to be restricted. The restricted sine function is restricted to \((-1, 1)\). \(\sin^{-1}x\) or arcsin(x), known as the inverse sine function, is the inverse of this restricted sine function.
02

Obtaining the Inverse

If we want to graph the inverse function, it can be obtained by reflecting the graph of the restricted sine function about the line \(y=x\). This is a general property of inverse functions.
03

Graphical Properties

The graph of arcsin(x) will rise from \(-\pi/2\) to \(\pi/2\) as x runs from -1 to 1. This is mirrored around the line \(y=x\), it will have a similar shape to the original sine curve 'on its side', starting low and rising up.

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