My book with answers to proakis and manolakis book says that $F(x,n)=x(-n + 2)$ is time invariant while my calculation shows that it is time variant. I have repeated them many times and I am mostly convinced that book is wrong. Can somebody please check my calculations?
First we shift output signal:
$$F(x, n - k) = x(-(n-k) + 2) = x(-n+k+2)$$
now lets put $x'(n) = x(n-k)$ and check what happen when we input shifted signal:
$$F(x', n) = x'(-n+2)=x(-n+2-k)=x(-n-k+2)$$
since $F(x, n - k) \neq F(x', n)$ this system is time variant. Is my method correct?