Mathematics

Problem

A function

g

is defined as

g(x)=\begin{cases} 2x-1, & x<1,\\ |x-1|+k, & 1\le x<3,\\ 4x+m, & x\ge 3, \end{cases}

where

k

and

m

are real constants. a. (i) Determine the value of

k

that ensures continuity of

g

x=1

. b. (ii) Using the value of

k

obtained, find the value of

m

that ensures continuity of

g

x=3

. c. (iii) With these constants, verify algebraically that

g

is continuous for all real

x