Mathematics

Problem

Consider the rational function

h:\mathbb R\setminus\{3\}\to\mathbb R

given by

h(x)=\dfrac{2x+5}{x-3}

. a. (i) Rewrite

h(x)

in the form

h(x)=2+\dfrac{11}{x-3}

and, using this form, discuss how the sign of

(x-3)

affects the behaviour of

h

on the intervals

(-\infty,3)

and

(3,\infty)

. b. (ii) Determine the range of

h

on each of the two intervals separately. c. (iii) Combine the results obtained in part (ii) to state the complete range of

h

on its domain

\mathbb R\setminus\{3\}