(a) Define the identity function on a non-empty set . (b) State the necessary and…

MATHEMATICS · Subjective · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

(a) Define the identity function on a non-empty set

A

. (b) State the necessary and sufficient condition that two functions

f:A\to B

and

g:B\to A

must satisfy in order to be inverses of each other. (c) Let

f:\mathbb R\to\mathbb R

be given by

f(x)=2x+3

and let

g:\mathbb R\to\mathbb R

be given by

g(x)=\dfrac{x-3}{2}

. Show that

f

and

g

are inverses of each other by explicitly evaluating both compositions

f\circ g

and

g\circ f

, and identifying the corresponding identity functions.