Q. 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.

It is given that f: {1, 2, 3} {a, b, c} given by

f(1) = a, f(2) = b and f(3) = c

So, if we define g:

{a,b,c} {1, 2, 3} as

g(a) = 1, g(b) = 2, g(c) = 3, then we get:

(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)) = g(a) = 1

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

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

Therefore, 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.

Then, 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,

So, if we define h: {1, 2, 3} {a, b, c} as

h(1) = a, h(2) = b, h(3) = c, then we get:

(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

goh = IX and hog = IY, where X = {1, 2, 3} and Y = {a, b, c}.

The inverse of g exists and g-1 = h

(f-1)-1 = h

h = f

(f-1)-1 = f

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functions - 0152 mins
Different kind of mappings58 mins
Range of Functions58 mins
Quick Revision of Types of Relations59 mins
Some standard real functions61 mins
Functions - 0947 mins
Quick Recap lecture of important graphs & functions58 mins