Question

If $A=\{1,2,3,4\}$, $B=\{3,5,7,9\}$, $C=\{7,23,47,79\}$ and $f:A\to B$, $f\left(x\right)=2x+1$, $g:B\to C$, $g\left(x\right)=x^2-2$ then write ${\left(gof\right)}^{-1}$ and $f^{-1}o{g}^{-1}$ in the form of ordered pair.

Answer 1

$A=\{1,2,3,4\}$, $B=\{3,5,7,9\}$, $C=\{7,23,47,79\}$

$f:A\to Bf\left(x\right)=2x+1$

$g:B\to C,g\left(x\right)=x^2-2$

Now, $gof\left(x\right)=gf\left(x\right)=g(2x+1)$

$=(2x+1{)}^2-2=4x^2+4x-1$

$\therefore gof\left(x\right)=4x^2+4x-1$

On putting $x=1,2,3,4$

$gof=\left\{\right(1,7),(2,23),(3,47),(4,79\left)\right\}$

$\because gof$is bijection function.

$\therefore$ Its inverse is possible

$\Rightarrow (gof{)}^{-1}=\left\{\right(7,1),(23,2),(47,3),(79,4\left)\right\}$

$\Rightarrow f^{-1}o{g}^{-1}=\left\{\right(7,1),(23,2),(47,3),(79,4\left)\right\}$

$\because (gof{)}^{-1}=f^{-1}o{g}^{-1}$ by theorem.