NCERT Solutions
Class 12 Maths
Relations and Functions
Ex. 1.3 Q11
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.
Function f: {1, 2, 3} → {a, b, c} is given by f(1) = a, f(2) = b, and f(3) = c
If we define g: {a, b, c} → {1, 2, 3} as g(a) = 1, g(b) = 2, g(c) = 3.
We have
(fog)(a) = f(g(a)) = f(1) = a
(fog)(b) = f(g(b)) = f(2) = b
(fog)(c) = f(g(c)) = f(3) = c
and
(gof)(1) = g(f(1)) = f(a) = 1
(gof)(2) = g(f(2)) = f(b) = 2
(gof)(3) = g(f(3)) = f(c) = 3
So, gof = IX and fog = IY, where X = {1, 2, 3} and Y= {a, b, c}.
Thus, the inverse of f exists and f−1 = g.
So, f−1 : {a, b, c} → {1, 2, 3} is given by f−1(a) = 1, f−1(b) = 2, f−1(c) = 3
Let us now find the inverse of f−1 i.e., find the inverse of g.
If we define h: {1, 2, 3} → {a, b, c} as h(1) = a, h(2) = b, h(3) = c
We have
(goh)(1) = g(h(1)) = g(a) = 1
(goh)(2) = g(h(2)) = g(b) = 2
(goh)(3) = g(h(3)) = g(c) = 3
and
(hog)(a) = h(g(a)) = h(1) = a
(hog)(b) = h(g(b)) = h(2) = b
(hog)(c) = h(g(c)) = h(3) = c
So, goh = IX and hog = IY, where X = {1, 2, 3} and Y = {a, b, c}.
Thus, the inverse of g exists and g−1 = h
⇒ (f−1)−1 = h
It can be noted that h = f
Hence, (f−1)−1 = f