State with reason whether following functions have inverse
(i) $f:\{ 1,\;2,\;3,\;4\} \to \{ 10\} \;\;with\;\;f = \{ (1,\;10),\;(2,\;10),\;(3,\;10),\;(4,\;10)\}$
(ii) $g:\{ 5,\;6,\;7,\;8\} \to \{ 1,\;2,\;3,\;4\} \;\;with\;\;g = \{ (5,\;4),\;(6,\;3),\;(7,\;4),\;(8,\;2)\}$
(iii) $h:\{ 2,\;3,\;4,\;5\} \to \{ 7,\;9,\;11,\;13\} \;\;with\;\;h = \{ (2,\;7),\;(3,\;9),\;(4,\;11),\;(5,\;13)\}$
(i) A function is invertible, if it is one-one and onto.
$f = \{ (1,\;10),\;(2,\;10),\;(3,\;10),\;(4,\;10)\}$
It is many-one function (see in figure(I)). Hence, f has noinverse,
(ii) g $= \{(5, 4), (6, 3), (7, 4), (8, 2)\}$
g is many-one fimction (see in figure (II)). Hence, g has noinverse.
(iii) h $= \{(2, 7), (3, 9), (4, 11), (5, 13)\}$
h is one-one and onto function, (see in figure (III))
Hence, h has an inverse.
NCERT & Exemplar solution for CBSE Class 12 Mathematics, Relations and Functions. Curated by Sachin Sharma. Free for all students.