If a function \(f : X \to Y\) defined as \(f(x) = y\) is one-one and onto, then we can define a unique function \(f(x) = y\) such that \(f(x) = y\), where \(f(x) = y\) and \(f(x) = y\), \(f(x) = y\). Function \(g\) is called the inverse of function \(f\).
The domain of 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. Let sine function be defined from set \(A\) to \([-1, 1]\) such that inverse of sine function exists, i.e., \(\sin^{-1} x\) is defined from \([-1, 1]\) to \(A\).
On the basis of the above information, answer the following questions:
(i) If \(A\) is the interval other than principal value branch, give an example of one such interval.
(ii) If \(\sin^{-1}(x)\) is defined from \([-1, 1]\) to its principal value branch, find the value of \(\sin^{-1}\left(-\frac{1}{2}\right) - \sin^{-1}(1)\).
(iii) Draw the graph of \(\sin^{-1} x\) from \([-1, 1]\) to its principal value branch.
(iv) Find the domain and range of \(f(x) = 2 \sin^{-1}(1 - x)\).
