Let f : A → B and g : B → C be the bijective functions

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) $f \circ g$

(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).

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