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.