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Question

# Let f : A → B and g : B → C be the bijective functions. Then, (gof)–1 = (a) f–1o g–1 (b) fog (c) g–1of–1 (d) gof

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Solution

## Given: f : A → B and g : B → C be the bijective functions $\mathrm{Since},f:A\to B\phantom{\rule{0ex}{0ex}}\mathrm{Thus},{f}^{-1}:B\to A...\left(1\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Since},g:B\to C\phantom{\rule{0ex}{0ex}}\mathrm{Thus},{g}^{-1}:C\to B...\left(2\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{From}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}{f}^{-1}o{g}^{-1}:C\to A...\left(3\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Also},gof:A\to C\phantom{\rule{0ex}{0ex}}⇒{\left(gof\right)}^{-1}:C\to A...\left(4\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Therefore},{\left(gof\right)}^{-1}={f}^{-1}o{g}^{-1}$ ​Hence, the correct option is (a). ​

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