# NCERT Class 12-Mathematics: Chapter –1 Relations and Functions Part 3

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**Question 13:**

Show that the function defined by , is neither one-one nor onto.

**Answer:**

For consider

We note that there are point, and with and , for instance, if we take and and , then we have and but . Hence is not one-one. Also, is not onto for if so then for such that which gives . But there is no such *x* in the domain , since the equation does not give any real value of .

**Question 14:**

Let be two functions defined as and . Then, find and .

**Answer:**

Here which can be redefined as

Similarly, the function *g* defined by may be redefined as

Therefore, *g o f* gets defined as :

For *x* ≥ 0, (*g o f* ) (*x*) = *g* (*f* (*x*) = *g* (2*x*) = 0

and for *x* < 0, (*g o f* ) (*x*) = *g* (*f* (*x*) = *g* (0) = 0.

Consequently, we have (*g o f* ) (*x*) = 0, ∀ *x* ∈ **R**.

Similarly, *f o g* gets defined as:

For *x* ≥ 0, (*f o g* ) (*x*) = *f* (*g* (*x*) = *f* (0) = 0,

and for *x* < 0, (*f o g* ) (*x*) = *f* (g(*x*)) = *f* (–2 *x*) = – 4*x*.

i.e.

**Question 15:**

Let be the set of real numbers and be the function defined by . Show that is invertible and find .

**Answer:**

Here the function is defined as (say). Then

This leads to a function defined as

Therefore,

Or

Similarly

Or .

Hence is invertible and which is given by

**Question 16:**

Let be a binary operation defined on **Q**. Find which of the following binary operations are associative

(i) .

(ii) for .

(iii) for .

(iv) for .

**Answer:**

(i) is not associative for if we take and , then

and

.

Thus and hence is not associative.

(ii) is associative since is associative with respect to multiplication.

(iii) is not associative for if we take and , then

, and

Thus and hence is not associative.

(iv) is not associative for if we take and , then

and .

Thus and hence is not associative.