Maths-
General
Easy

Question

# Let  denote the set of all integers and  defined by is

Hint:

## The correct answer is: onto but not one-one

### The given function isf(x) =             if x is even= 0                if x is oddThe 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) = 0For even values of x  f(x) = So, if x = 2           f(x) =  = 1If x = 4            f(x) =  = 2If x = -2            f(x) =  = -1If x = - 4         f(x) =  = -2So, 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.