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largest domain of the function

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Consider the function f(x) = arcsin(1 - x^2).
(a) What is the largest domain over which this function is defined?
(b) For this domain, what is the corresponding range of the function?
(c) If this function is one-to-one, find the inverse function f^-1(x), and state the domain of this inverse function. Otherwise, find a suitable restriction for the domain of the function, so that the restricted function has an inverse. Find this inverse function,carefully specifying its domain and range.

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Solution Summary

In this solution, we clearly assess the largest domain of the given function.

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Let y = f(x) = arcsin(1 - x^2), then 1 - x^2 = siny and we know |siny|<=1 for any y
(a) We have |1 - x^2| <= 1 and thus 0<= x^2 <= 2 and thus -sqrt(2) <= x <= sqrt(2)
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