Let and be two functions

MATHEMATICS · Multi-part · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

Let

f:A\to B

and

g:B\to A

be two functions. Answer the following questions. (i) State the two composition identities that must hold when

g

is the inverse of

f

. Write each identity using function notation. (ii) Give a logical explanation of why both identities stated in part (i) are necessary to conclude that

g

is indeed the inverse of

f

. Your explanation should refer to the meanings of domain and range. (iii) Construct explicit functions

f

and

g

(specify their domains and codomains) such that

f\bigl(g(x)\bigr)=x

for every

x

in the domain of

g

, but

g\bigl(f(x)\bigr)\ne x

for at least one

x

in the domain of

f

. Show all working to justify your claim.

Related theory

Composition of Functions
Functions
Exponential and Logarithmic Functions
Composite functions
Relations and Functions
Invertible functions and inverse of a function

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