Let $f : A \to B a n d g : B \to A$ be two functions such that $g o f = I_{A} a n f f o g = I_{B} .$ Then, which of the following statement/s is/are true?

{(f^{- 1})}^{- 1} = f

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{(g o f)}^{- 1} = f^{- 1} o g^{- 1}

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f and g are bijections

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g = f^{- 1}

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Solution

The correct options are
A ${(f^{- 1})}^{- 1} = f$
B ${(g o f)}^{- 1} = f^{- 1} o g^{- 1}$
C f and g are bijections
D $g = f^{- 1}$
$Given gof = I_{A} and fog = I_{B} . \Rightarrow f and g are inverse of each other . \Rightarrow f and g are bijections and Inverse of f and g exist . \Rightarrow g = f^{- 1} Option 1 and 2 obviously hold by the properties of functions .$

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