Maths-

General

Easy

### Question

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

- onto but not one-one
- one-one but onto
- one-one and onto
- neither one-one nor onto

### Hint:

We are given a function f(x). The domain and co-domain of the function are set of all integers. We have to find if the given function is one-one and onto.

#### 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.

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