Let f : X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g 1 and g 2 are two inverses of f . Then for all y ∈ Y , f o g 1 ( y ) = I Y ( y ) = f o g 2 ( y ). Use one-one ness of f ).

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Solution

Let a function f:X→Y be an invertible function and let the function f possess two inverses that are g 1 and g 2 .

The equation for the condition of invertible function is,

fo g 1 ( y )=fo g 2 ( y ) f( g 1 ( y ) )=f( g 2 ( y ) ) g 1 ( y )= g 2 ( y ) [ ∵Since f(x) is one-one ] g 1 = g 2

Therefore, the value of the function g is one-one.

Thus, the function f has a unique inverse.

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