Question

State with reason whether following functions have inverse
(i) $f:\{1,2,3,4\}\to \left\{10\right\}$ with
$f=\left\{\right(1,10),(2,10),(3,10),(4,10\left)\right\}$
(ii) $g:\{5,6,7,8\}\to \{1,2,3,4\}$ with
$g=\left\{\right(5,4),(6,3),(7,4),(8,2\left)\right\}$
(iii) $h:\{2,3,4,5\}\to \{7,9,11,13\}$ with
$h=\left\{\right(2,7),(3,9),(4,11),(5,13\left)\right\}$

Answer 1

$\left(i\right)A$ function has inverse if it is one-one and on-to.

$f=\left\{\right(1,10),(2,10),(3,10),(4,10\left)\right\}$

Since all elements have image $10$, they do not have unique image.

$\therefore$ f is not one-one.

Since, f is not one-one,it do not have an inverse.

$\left(ii\right)g=\left\{\right(5,4),(6,3),(7,4),(8,2\left)\right\}$

Since,$5$ and $7$ have same image $4$,g is not one-one.

Since g is not one-one,it does not have on inverse.

$\left(iii\right)h=\left\{\right(2,7),(3,9),(4,11),(5,13\left)\right\}$

Since each element has a unique image h is one-one.

since,for every element,there is a corresponding element, $\therefore$ his onto

Since function is both one--one and onto,it will have inverse

$h=\left\{\right(2,7),(3,9),(4,11),(5,13\left)\right\}{h}^{-1}=\left\{\right(7,2),(9,3),(11,4),(13,5\left)\right\}$