Mathematics

CBSE Class 12

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2.

Show that f is invertible and find the inverse of f. Here, W is the set of all whole numbers.

Show that f is invertible and find the inverse of f. Here, W is the set of all whole numbers.

Let f: W→W be defined as

We need to prove that 'f' is invertible.

In order to prove that 'f' is invertible it is sufficient to prove that f is a bijection.

A function f: A→B is a one-one function or an injection, if

f(x) = f(y) ⇒ x = y for all x, y ∈ A.

Case i:

If x and y are odd.

Let f(x) = f(y)

⇒x − 1 = y − 1

⇒x = y

Case ii:

If x and y are even,

Let f(x) = f(y)

⇒x + 1 = y + 1

⇒x = y

Thus, in both the cases, we have,

f(x) = f(y) ⇒ x = y for all x, y ∈ W.

Hence f is an injection.

Let n be an arbitrary element of W.

If n is an odd whole number, there exists an even whole number n − 1 ∈ W such that

f(n − 1) = n − 1 + 1 = n.

If n is an even whole number, then there exists an odd whole number n + 1 ∈ W such that f(n + 1) = n + 1 − 1 = n. Also, f(1) = 0 and f(0) = 1

Thus, every element of W (co-domain) has its pre-image in W (domain).

So f is an onto function.

Thus, it is proved that f is an invertible function.

Thus, a function g: B→A which associates each element y ∈ B to a unique element x ∈ A such that f(x) = y is called the inverse of f.

That is, f(x) = y ⇔ g(y) = x.

The inverse of f is generally denoted by f ^{-1}.

Now let us find the inverse of f.

Let x, y ∈ W such that f(x) = y

⇒x + 1 = y, if x is even

And

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6.

Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary operation on A defined by (a, b) * (c, d) = (ac, b+ad) for (a, b), (c, d) A. Then find

(i) The identify element of * in A.

(ii) Invertible elements of A, and write the inverse of elements (5, 3) and

Let A = Q x Q, where Q is the set of rational numbers.

Given that * is the binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for

(a, b), (c, d) ∈ A.

(i)

We need to find the identity element of the operation * in A.

Let (x, y) be the identity element in A.

Thus,

(a, b) * (x, y) = (x, y) * (a, b) = (a, b), for all (a, b) ∈ A

⇒(ax, b + ay) = (a, b)

⇒ ax = a and b + ay =b

⇒ y = 0 and x = 1

Therefore, (1, 0) ∈ A is the identity element in A with respect to the operation *.

(ii) We need to find the invertible elements of A.

Let (p, q) be the inverse of the element (a, b)

Thus,

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8.

Find the absolute maximum and absolute minimum values of the function f given by

Of these values, the maximum value is 1, and the minimum value is −1.

Thus, the absolute maximum and absolute minimum values of f(x) are 1 and −1, which it attains at x = 0 and

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Write the element of the matrix whose elements are given by

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