Study the following assertions about functions and give complete justifications.

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Study the following assertions about functions and give complete justifications. (i) Statement

S_{1}

: “If

f,g:\mathbb{R}\to\mathbb{R}

satisfy

f\circ I_{\mathbb{R}}=g

, then

f=g

.” Decide whether

S_{1}

is true or false and justify your conclusion. (ii) Statement

S_{2}

: “If functions

f:A\to B

and

h:B\to A

satisfy

h\circ f=I_{A}

, then

f

is necessarily invertible.” Prove or disprove

S_{2}

. (iii) If your answer to (ii) is affirmative, show that

h=f^{-1}

and prove both compositions with the appropriate identity functions. (iv) Provide an explicit example of non-empty sets

A,B

and functions

f:A\to B

h:B\to A

with

f\circ h=I_{B}

but

h\circ f\neq I_{A}

. Explain why neither

f

nor

h

is invertible in your example.