about to only mathematics

Answer

`f^(-1)=y`

Explanation

It gives three cases
Case I When f(x) = 1 is true.
In this case, remaining two are false.
`therefore f(y) =1 and f(z) = 2`
This means x and y have the same image, so f(x) is not an injective, which is a contradiction.
Case II When `f(y) ne 1` is true.
If `f(y) ne 1` is true, then the remaining statements are false.
`therefore f(x) ne 1 and f(z)=2`
i.e. both x and y are not mapped to 1. So, either both associate to 2 or 3. Thus, it is not injective.
Case III When `f(z)ne 2` is true.
If `f(z) ne 2` is true, then remaining statements ar false.
`therefore " If " f(x) ne 1 and f(y) =1`
But f is Injective.
Thus, we have `f(x)=2, f(y)=1 and f(z)=3`
Hence, `f^(-1)(1)=y`