Question:
Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be the bijective functions. Then, $(g o f)^{-1}=$
(a) $f^{-1} \circ g^{-1}$
(b) fog
(c) $g^{-1} o r^{-1}$
(d) gof
Solution:
Given: f : A → B and g : B → C be the bijective functions
Since, $f: A \rightarrow B$
Thus, $f^{-1}: B \rightarrow A$ $\ldots(1)$
Since, $g: B \rightarrow C$
Thus, $g^{-1}: C \rightarrow B$ $\ldots(2)$
From (1) and (2), we get
$f^{-1} \mathrm{og}^{-1}: C \rightarrow A$ $\ldots(3)$
Also, $g o f: A \rightarrow C$
$\Rightarrow(g o f)^{-1}: C \rightarrow A$ $\ldots(4)$
Therefore, $(g o f)^{-1}=f^{-1} \mathrm{og}^{-1}$
Hence, the correct option is (a).