NCERT Class 12-Mathematics: Chapter – 1 Relations and Functions Part 3 (For CBSE, ICSE, IAS, NET, NRA 2023)

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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 = g = 0

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

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

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.