Class 12 Maths

Relation and Function

NCERT Exemplar Problem

Long Answer Type Part 2

Question 20: Let A = R – {3}, B = R – {1}. Let NCERT Exemplar Problems and Solution class 12 Math (29) be defined by NCERT Exemplar Problems and Solution class 12 Math (30) Then show that f is bijective.

Solution: Given,

NCERT Exemplar Problems and Solution class 12 Math (31)

Now, for injectivity:

NCERT Exemplar Problems and Solution class 12 Math (32)

After cross multiplication, we get

NCERT Exemplar Problems and Solution class 12 Math (33)

Thus, f(x) is an injective function.


Now, for surjectivity:

NCERT Exemplar Problems and Solution class 12 Math (34)

Therefore, f(x) is a surjective function.

Here, we can see that f(x) is a surjective and injective both funtion.

Thus, f(x) is bijective.

Question 21: Let A = [– 1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

NCERT Exemplar Problems and Solution class 12 Math (35)

Solution:

NCERT Exemplar Problems and Solution class 12 Math (36)

This shows that f(x) is one-one

NCERT Exemplar Problems and Solution class 12 Math (37)

Clearly, f(x) is not onto.

Thus, f(x) is not bijective as it is one-one and not onto.

NCERT Exemplar Problems and Solution class 12 Math (38)

Solution:

NCERT Exemplar Problems and Solution class 12 Math (39)

Clearly, g(x) is not one-one

NCERT Exemplar Problems and Solution class 12 Math (40)

Here, it is also clear that g(x) is not onto.

Since, g(x) is neither one-one nor onto, thus g(x) is not bijective.



NCERT Exemplar Problems and Solution class 12 Math (41)

Solution:

NCERT Exemplar Problems and Solution class 12 Math (42)

Hence, h(x) is a surjective function.

There h(x) is bijective.

NCERT Exemplar Problems and Solution class 12 Math (43)

Solution:

NCERT Exemplar Problems and Solution class 12 Math (44)

Here, it is clear that k(x) is not one-one.

NCERT Exemplar Problems and Solution class 12 Math (45)

Exercise 1

Exemplar Problems