# How do you graph y=sin^-1(x-2) over the interval 1<=x<=3?

Dec 22, 2017

#### Explanation:

As $y = {\sin}^{- 1} x$ means $x = \sin y$, the graph is similar to graph of $y = \sin x$ but wave is formed along $y$-axis. The range of $y = {\sin}^{- 1} x$ is, however, $\left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$ and hence its graph is limited between $\left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$ along $y$-axis and corresponding values of $x$ ranges from $\left[- 1 , 1\right]$,

other values are $\left(- 1 , - \frac{\pi}{2}\right) , \left(- \frac{\sqrt{3}}{2} , - \frac{\pi}{3}\right) , \left(- \frac{1}{\sqrt{2}} , - \frac{\pi}{4}\right) , \left(- \frac{1}{2} , - \frac{\pi}{6}\right) , \left(0 , 0\right) , \left(\frac{1}{2} , \frac{\pi}{6}\right) , \left(\frac{1}{\sqrt{2}} , \frac{\pi}{4}\right) , \left(\frac{\sqrt{3}}{2} , \frac{\pi}{3}\right) , \left(1 , \frac{\pi}{2}\right)$.

The graph appears as follows:

graph{arcsinx [-10, 10, -5, 5]}

However, in $y = {\sin}^{- 1} \left(x - 2\right)$ $x$ can take values from $\left[1 , 3\right]$ and hence the graph is similar to that of $y = {\sin}^{- 1} x$ but shifted $2$ units to right. The graph appears as follows:

graph{arcsin(x-2) [-10, 10, -5, 5]}