Mathematics

Problem

Given two real-valued functions

f

and

g

defined by

f(x)=\dfrac{1}{x-3},\qquad g(x)=\sqrt{5-x}.

Answer the following. a. (i) State the largest subset of

\mathbb{R}

on which both

f

and

g

are simultaneously defined. Give a reason for every value that is excluded. b. (ii) For the quotient function

h(x)=\dfrac{f(x)}{g(x)}

, decide whether any further restriction on

x

is necessary apart from what you obtained in part (i). Justify your answer and write the complete domain of

h

in set–builder form.