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

Solution:

A function is a process or a relation that associates each element 'a' of a non-empty set A, to a single element 'b' of another non-empty set B.

According to the given problem,

Let f : X → Y be an invertible function.

In general, a function is invertible as long as each input features a unique output. That is, every output is paired with exactly one input

Then there exists a function

g : Y → X such that

gof = I_X and fog = I_Y

Here, f ^-1 = g

Now,

gof = I_X and fog = I_Y

⇒ f ^-1of = I and fof ^-1 = I

Hence,

f ^-1 : Y → X is invertible

and f ^-1 is f i.e., (f ^{- 1})^{- 1} = f

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.3 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.

Summary:

Given that f : X → Y be an invertible function. Hence we have shown that  the inverse of f^{- 1} is f i.e., (f^{- 1})^{- 1} = f