If a function \( f : X \to Y \) defined as \( f(x) = y \) is one-one and onto, then we can define a unique function \( g : Y \to X \) such that \( g(y) = x \), where \( x \in X \) and \( y = f(x) \), \( y \in Y \). Function \( g \) is called the inverse of function \( f \).
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The domain of the sine function is \( \mathbb{R} \) and function sine: \( \mathbb{R} \to \mathbb{R} \) is neither one-one nor onto. The following graph shows the sine function.
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Let sine function be defined from set \( A \) to \( [-1, 1] \) such that the inverse of the sine function exists, i.e., \( \sin^{-1} x \) is defined from \( [-1, 1] \) to \( A \).
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On the basis of the above information, answer the following questions:
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If \( A \) is the interval other than the principal value branch, give an example of one such interval.
If \( \sin^{-1}(x) \) is defined from \( [-1, 1] \) to its principal value branch, find the value of \( \sin^{-1}\left(-\frac{1}{2}\right) - \sin^{-1}(1) \).
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Draw the graph of \( \sin^{-1} x \) from \( [-1, 1] \) to its principal value branch.
OR Find the domain and range of \( f(x) = 2 \sin^{-1}(1 - x) \).
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