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# 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 belongs to Y, fog 1 (y) = 1 Y (y) = fog 2 (y). Use one-one ness of f).

Q.10 Let $f : X \rightarrow Y$be an invertible function. Show that f has a unique inverse.
(Hint: suppose $g_1$ and $g_2$ are two inverses of $f$. Then for all $y \in Y$,
$fog_1 (y) = I_Y (y) = fog_2 (y)$. Use one-one ness of f).

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Let $f : X \rightarrow Y$be an invertible function

Also, suppose f has two inverse $g_1 and g_2$

For $y \in Y$, we have

$fog_1(y) = I_y(y)=fog_2(y)$

$\Rightarrow$     $f(g_1(y))=f(g_2(y))$

$\Rightarrow$             $g_1(y)=g_2(y)$                     [f is invertible implies f is one - one]

$\Rightarrow$                     $g_1=g_2$                        [g is one-one]

Thus,f has a unique inverse.

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