If a function f: X → Y defined as f ( x )= y is one-one and onto, then we can define a unique function g: Y → X such that g( y )= x, where x ∈ X and…

If a function $\displaystyle \mathrm{f}: \mathrm{X} \rightarrow \mathrm{Y}$ defined as $\displaystyle \mathrm{f}(\mathrm{x})=\mathrm{y}$ is one-one and onto, then we can define a unique function $\displaystyle g: \mathrm{Y} \rightarrow \mathrm{X}$ such that $\displaystyle g(\mathrm{y})=\mathrm{x}$, where $\displaystyle \mathrm{x} \in \mathrm{X}$ and $\displaystyle \mathrm{y}=\mathrm{f}(\mathrm{x}), \mathrm{y} \in \mathrm{Y}$. Function $\displaystyle g$ is called the inverse of function $\displaystyle \mathrm{f}$. The domain of sine function is $\displaystyle R$ and function sine $\displaystyle : R \rightarrow R$ is neither one-one nor onto. The following graph shows the sine function.

Let sine function be defined from set A to $\displaystyle [-1,1]$ such that inverse of sine function exists, i.e., $\displaystyle \sin ^{-1} \mathrm{x}$ is defined from $\displaystyle [-1,1]$ to A . On the basis of the above information, answer the following questions :

If A is the interval other than principal value branch, give an example of one such interval.

If $\displaystyle \sin ^{-1}(\mathrm{x})$ is defined from [-$\displaystyle 1,1$] to its principal value branch, find the value of $\displaystyle \sin ^{-1}\left(-\frac{1}{2}\right)-\sin ^{-1}(1)$.