Question

Let $f:A\to Bandg:B\to A$be two functions such that $gof=I_{A}anffog=I_B.$ Then, which of the following statement/s is/are true?

• A. ${\left(f^{-1}\right)}^{-1}=f$
• B. ${\left(gof\right)}^{-1}=f^{-1}o{g}^{-1}$
• C. f and g are bijections
• D. $g=f^{-1}$

Answer 1

Answer: (A), (B), (C), (D)

$g=f^{-1}$

$\mathrm{Given}\mathrm{gof}=I_A\mathrm{and}\mathrm{fog}=I_B.\Rightarrow f\mathrm{and}g\mathrm{are}\mathrm{inverse}\mathrm{of}\mathrm{each}\mathrm{other}.\Rightarrow f\mathrm{and}g\mathrm{are}\mathrm{bijections}\mathrm{and}\mathrm{Inverse}\mathrm{of}f\mathrm{and}g\mathrm{exist}.\Rightarrow g=f^{-1}\mathrm{Option}1\mathrm{and}2\mathrm{obviously}\mathrm{hold}\mathrm{by}\mathrm{the}\mathrm{properties}\mathrm{of}\mathrm{functions}.$