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1.Relation and Function
medium
Let $f: X \rightarrow Y$ be an invertible function. Show that the inverse of $f^{-1}$ is $f$, i.e., $\left(f^{-1}\right)^{-1}=f$.
Option A
Option B
Option C
Option D
Solution
Let $f : X \rightarrow Y$ be an invertible function.
Then, there exists a function $g : Y \rightarrow X$ such that $gof = I_X$ and $fog = I_Y$
Here, $f^{-1}=g$
Now,
$gof = I_X$ and $fog = I _{ Y }$
$\Rightarrow f ^{-1}$ of $= I_X$ and $fof -1= I_Y$
Hence, $f^{-1}: Y \rightarrow X$ is invertible and $f$ is the inverse of $f^{-1}$ i.e., $\left(f^{-1}\right)^{-1}=f$
Standard 12
Mathematics
