NCERT Solutions for CBSE Class 12 Mathematics ‒ Chapter 1: Relations and Functions (Part V)

NCERT Solutions for CBSE Class 12 Maths, Chapter 1: Relations & Functions are available in this article. Here, you will get solutions to the questions of exercise 1.3 from question number 8 to question number 14. These questions are related to composition of functions and invertible function.

NCERT Solutions for CBSE Class 12th Maths, Chapter 1: Relations and Functions (Exercise 1.3) are given below

Question8:

Solution 8:

NCERT Exemplar Class 12 Mathematics – Chapter 1 Relations and Functions

Question 9:

Solution 9:

NCERT Solutions for CBSE Class 12 Physics: All Chapters

Question 10: Let f : X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g₁ and g₂ are two inverses of f. Then for all y ∈ Y, fog₁(y) = I_Y(y) = fog₂(y). Use one-one ness of f).

Solution 10:

Let Y is an invertible function

Suppose g₁ & g₂ are two inverses of f

Now, for all y ϵ Y,

fog₁ (y) = fog₂ (y)

f[g₁(y)] = f[g₂(y)]

g₁(y) = g₂(y) {f is one-one}

g₁ = g₂ (g is one-one)

Question 11: Consider f : {1, 2, 3} → {a, b, c} given by f (1) = a, f (2) = b and f (3) = c. Find f ^–1 and show that (f ^–1)^–1 = f.

Solution 11:

f: {1, 2, 3} → {a, b, c}

f(1) = a 

f(2) = b 

 f(3) = c

Let g: {a, b, c} → {1, 2, 3}

g(a) = 1

g(b) = 2

g(c) = 3

Now,

(fog) (a) = f [g(a)] = f (1) = a

(fog) (b) = f [g(b)] = f(2) = b

(fog) (c) = f [g(c)] = f (3) = c

(gof) (1) = g [f(1)] = g(a) =1

(gof) (2) = g [f(2)] = g(b) =2

(gof) (3) = g [f(3)] = g(c) =3

Thus, gof = fog

gof  = I_X

fog = I_Y

Where X = {1, 2, 3} and Y= {a, b, c}

Thus, f⁻¹ = g

Hence, f⁻¹: {a, b, c} → {1, 2, 3}

f⁻¹(a) = 1

f⁻¹(b) = 2

 f^-1(c) = 3

Let us calculate the inverse of f⁻¹ 

h: {1, 2, 3} → {a, b, c}

h(1) = a, h(2) = b, h(3) = c

Then,

(goh) (1) = g[h(1)] = g(a) = 1

(goh) (2) = g[h(2)] = g(b) = 2

(goh) (3) = g[h(3)] = g(c) = 3

(hog)(a) = h [g(a)]= h (1) = a

(hog)(b) = h [g(b)]= h (2) = b

(hog)(c) = h [g(c)]= h (3) = c

goh = I_X

hog = I_Y

Now,

g⁻¹ = h 

(f⁻¹)⁻¹ = h

We can observe that

 h = f

Hence, (f^‒1)^‒1 = f.

Question 12: Let f : X → Y be an invertible function. Show that the inverse of f^–1 is f, i.e., (f ^–1)^–1 = f.

Solution 12:

Question 13: If f : R → R be given by f (x) = (3 − x³ )^1/3 , then fof (x) is

(A) x³

(B) x³

(C) x

(D) (3 – x³).

Solution 13:

Question 14:

Solution 14:

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