Given:
The given function is:
[tex]f(x)=11x-11[/tex]
To find:
The value of [tex](f\circ f^{-1})(-11)[/tex].
Solution:
We have,
[tex]f(x)=11x-11[/tex]
First we have to find the inverse of this function.
Putting f(x)=y, we get
[tex]y=11x-11[/tex]
Interchange x and y.
[tex]x=11y-11[/tex]
Isolate the variable y.
[tex]x+11=11y[/tex]
[tex]\dfrac{x+11}{11}=y[/tex]
[tex]y=\dfrac{x+11}{11}[/tex]
Putting [tex]y=f^{-1}(x)[/tex], we get
[tex]f^{-1}(x)=\dfrac{x+11}{11}[/tex]
Now,
[tex](f\circ f^{-1})(-11)=f(f^{-1}(-11))[/tex]
[tex](f\circ f^{-1})(-11)=f\left(\dfrac{-11+11}{11}\right)[/tex] [tex]\left[\because f^{-1}(x)=\dfrac{x+11}{11}\right][/tex]
[tex](f\circ f^{-1})(-11)=f\left(\dfrac{0}{11}\right)[/tex]
[tex](f\circ f^{-1})(-11)=f\left(0\right)[/tex]
The given function is [tex]f(x)=11x-11[/tex].
[tex](f\circ f^{-1})(-11)=11(0)-11[/tex]
[tex](f\circ f^{-1})(-11)=-11[/tex]
Therefore, the value of [tex](f\circ f^{-1})(-11)[/tex] is [tex]-11[/tex]. Hence the correct option is A.
