To find the inverse of the function [tex]\( y = \cos^{-1}(x - \pi) \)[/tex], we need to follow a systematic approach:
1. Understand the original function:
[tex]\[
y = \cos^{-1}(x - \pi)
\][/tex]
This function represents an inverse cosine function.
2. Switch the variables to find the inverse function:
[tex]\[
x = \cos^{-1}(y - \pi)
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex], we can follow these steps:
- Apply the cosine function on both sides:
[tex]\[
\cos(x) = y - \pi
\][/tex]
- Rearrange to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \pi + \cos(x)
\][/tex]
4. Identify the correct option:
We need to compare the final expression [tex]\( y = \pi + \cos(x) \)[/tex] with the provided choices:
- a. [tex]\( y = \pi - \cos(x) \)[/tex]
- b. [tex]\( y = \cos(x - \pi) \)[/tex]
- c. [tex]\( y = \pi + \cos(x) \)[/tex]
- d. [tex]\( y = \cos(x + \pi) \)[/tex]
The correct match is:
[tex]\[
y = \pi + \cos(x)
\][/tex]
Therefore, the best answer from the choices provided is:
C
