To find the inverse of the function [tex]\( y = \frac{\pi}{4} + \sin x \)[/tex], let's follow the steps outlined below:
1. Start with the original function:
[tex]\[
y = \frac{\pi}{4} + \sin x
\][/tex]
2. Swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex] to begin finding the inverse:
[tex]\[
x = \frac{\pi}{4} + \sin y
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
- First, isolate [tex]\( \sin y \)[/tex] by subtracting [tex]\(\frac{\pi}{4}\)[/tex] from both sides:
[tex]\[
x - \frac{\pi}{4} = \sin y
\][/tex]
- Next, apply the arcsine (inverse sine) function to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \arcsin\left(x - \frac{\pi}{4}\right)
\][/tex]
Hence, the inverse function is:
[tex]\[
y = \arcsin\left(x - \frac{\pi}{4}\right)
\][/tex]
Examining the provided options, the correct choice is:
[tex]\[
\boxed{\text{c. } y = \operatorname{Arcsin}\left(x - \frac{\pi}{4}\right)}
\][/tex]
