# NCERT Class 12-Mathematics: Chapter – 1 Relations and Functions Part 9 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 19**:

Give an example of a map

(i) Which is one-one but not onto

(ii) Which is not one-one but onto

(iii) Which is neither one-one nor onto.

**Answer**:

(i) Which is one-one but not onto

Let , be a function given by .

In order to prove that f is one-one, it is sufficient to prove that

Now, let

is one-one.

is not onto, as for , there does not exist any in N such that .

Thus, , be a function given by , which is one-one but not onto.

(ii) Which is not one-one but onto

Let , be a function given by and for every .

Since,

and does not have unique image.

Thus, f is not one-one.

Let

for each there exists such that .

Thus, f is onto.

(iii) Which is neither one-one nor onto.

Let the function be defined by .

We have,

In order to prove that f is one-one, it is sufficient to prove that .

Let and are two different elements in R.

Now,

We observe that but _{.}

This shows that different element in R may have same image.

Thus, is not one-one.

We know that lies between and .

So, the range of f is which is not equal to its co-domain.

i.e.. , range of (co-domain)

In other words, range of f is less than co-domain, i.e.. , there are elements in co-domain which does not have any pre-image in domain.

So, f is not onto.

Hence, f is neither one-one nor onto.

**Question 20**:

Let be defined by Then show that f is bijective.

**Answer**:

Given that,

In order to prove that f is one-one, it is sufficient to prove that , .

Let

is one-one.

f is onto if every element of B is the f-image of some element of A.

Thus, for each y ∈ B there exists such that .

Hence, f is onto.

Since, f is one-one and onto therefore f is bijective.