Restrict the domain to take this inverse function.
A function can only be an inverse if it is 1-to-1 and undoes exactly the desired function. See inverse function notes for a review of inverse functions. In the graph at the left, notice that the sine function, pink and dashed, is not 1-to-1 because it is periodic and repeats every 2 . The only way for f(x) = sin(x) to undo g(x)= sin^{-1}(x) is if it is 1-to-1 which requires the domain to be restricted. Once this is done f(x) = sin(x) undoes g(x)= sin^{-1}(x) and g(x)= sin^{-1}(x) undoes f(x) = sin(x). If we only use the pink part, from - to + , then when the restricted sine is reflected over the line y=x to take the inverse graphically, the inverse, g(x) = sin^{-1}(x) is found and will indeed give us a single value for every x value from -1 to 1, inclusive. This restriction makes the domain of the Arcsine - < x <+ and the range to -1< y < 1 as is needed. |
When you solve an equation, be mindful of the domain of the x in the equation and how many solutions you should be looking for.
Though the Arcsine function is used to solve the equation, solutions to the equation may not be in the range of the arcsine function. |