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Question
Let f: A → B and g: B → C be the bijective functions. Then (g o f)–1 is ______.
Options
f –1 o g–1
f o g
g–1 o f–1
g o f
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Solution
Let f: A → B and g: B → C be the bijective functions. Then (g o f)–1 is f –1 o g–1.
Explanation:
Given that, f: A → B and g: B → C be the bijective functions.
(f –1 o g–1) o (g o f) = f –1 o (g–1 o g o f)
= f –1 o (g–1 o g) o f ......(As composition of functions is associative)
= f –1 o IB o f) .......(Where IB is identity function on B)
= (f –1 o IB) o f
= f –1 o f
= IA
Thus (g o f)–1 = f –1 o g –1
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