# Relation and Function

## NCERT Exemplar Problem

### Long Answer Type Part 2

Question 20: Let A = R – {3}, B = R – {1}. Let be defined by Then show that f is bijective.

**Solution:** Given,

Now, for injectivity:

After cross multiplication, we get

Thus, f(x) is an injective function.

Now, for surjectivity:

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:

**Solution:**

This shows that f(x) is one-one

Clearly, f(x) is not onto.

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

**Solution:**

Clearly, g(x) is not one-one

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.

**Solution:**

Hence, h(x) is a surjective function.

There h(x) is bijective.

**Solution:**

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