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Write the equation for the inverse of the function:

[tex]\[ y = \operatorname{Cos}^{-1}(x - \pi) \][/tex]

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]

Please select the best answer from the choices provided:

A, B, C, or D.



Answer :

GinnyAnswer
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

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