Let  denote the set of all integers and  defined by is

The correct answer is: onto but not one-one

The given function is
f(x) =             if x is even
= 0                if x is odd
The value of function will be  when the we substitute even integers for x. When we substitute odd Integers in place of x we get the value 0.
We have to check if the given function is one-one or/and onto.
1) One-One function: A function is said to be one-one function if for every element in domain there is unique element in co-domain.
For this function, every odd number has same image in the co-domain.
So, the function is not one-one.
2) onto function: A function is said to be onto if every element in co-domain have some pre image.  It means the domain and range of the function should be equal.
For this function, co-domain is set of all integers.
Let's find the range.
For odd values of x  f(x) = 0
For even values of x  f(x) = 
So, if x = 2           f(x) =  = 1
If x = 4            f(x) =  = 2
If x = -2            f(x) =  = -1
If x = - 4         f(x) =  = -2
So, range ={...-2, -1, 0, 1, 2, ...}
Range is set of all integers.
Hence co-domain and range are equal.
So, the given function is onto.
The given function is onto but not one-one.

The letter Z represents set of Integers. We should know definition of different types of functions to solve such questions.