Answer:
C. [tex]f(x)=\frac{x+1}{x-1}[/tex]
Step-by-step explanation:
Let's find the inverse of each of the given options.
Option A:
[tex]f(x)=\frac{x+6}{x-6}\\y=\frac{x+6}{x-6}[/tex]
To find [tex]f^{-1}(x)[/tex], replace 'x' with 'y' and 'y' with 'x'. This gives,
[tex]x=\frac{y+6}{y-6}[/tex]
Rewrite in terms of 'y'. This gives,
[tex]x(y-6)=y+6\\xy-6x=y+6\\xy-y=6x+6\\y=\frac{6x+6}{x-1}[/tex]
The given function [tex]y=\frac{6x+6}{x-1}\ne y=\frac{x+6}{x-6}[/tex]
So, option A is incorrect.
Option B:
[tex]f(x)=\frac{x+2}{x-2}\\y=\frac{x+2}{x-2}[/tex]
To find [tex]f^{-1}(x)[/tex], replace 'x' with 'y' and 'y' with 'x'. This gives,
[tex]x=\frac{y+2}{y-2}[/tex]
Rewrite in terms of 'y'. This gives,
[tex]x(y-2)=y+2\\xy-2x=y+2\\xy-y=2x+2\\y=\frac{2x+2}{x-1}[/tex]
The given function [tex]y=\frac{2x+2}{x-1}\ne y=\frac{x+2}{x-2}[/tex]
So, option B is incorrect.
Option C:
[tex]f(x)=\frac{x+1}{x-1}\\y=\frac{x+1}{x-1}[/tex]
To find [tex]f^{-1}(x)[/tex], replace 'x' with 'y' and 'y' with 'x'. This gives,
[tex]x=\frac{y+1}{y-1}[/tex]
Rewrite in terms of 'y'. This gives,
[tex]x(y-1)=y+1\\xy-x=y+1\\xy-y=x+1\\y=\frac{x+1}{x-1}[/tex]
The given function [tex]y=\frac{x+1}{x-1}\ equals\ y=\frac{x+1}{x-1}[/tex]
So, option C is correct.
Option D:
[tex]f(x)=\frac{x+5}{x-5}\\y=\frac{x+5}{x-5}[/tex]
To find [tex]f^{-1}(x)[/tex], replace 'x' with 'y' and 'y' with 'x'. This gives,
[tex]x=\frac{y+5}{y-5}[/tex]
Rewrite in terms of 'y'. This gives,
[tex]x(y-5)=y+5\\xy-5x=y+5\\xy-y=5x+5\\y=\frac{5x+5}{x-1}[/tex]
The given function [tex]y=\frac{5x+5}{x-1}\ne y=\frac{x+6}{x-6}[/tex]
So, option D is incorrect.
Therefore, only option C is correct.
