Mathematics

Problem

A piecewise–defined function

g

is given by

g(x)=\begin{cases} ax+b,& x<1,\\ x^2,& 1\le x<3,\\ \ln (cx),& x\ge 3, \end{cases}

where

a,\,b,

and

c

are real constants. The function

g

is known to be continuous for all real numbers. a. Part 1: (a) Write the two distinct equations that the constants

a,\,b,

and

c

must satisfy to ensure continuity of

g

x=1

and

x=3

, respectively. b. Part 2: (b) Solve the system obtained in part (a) to determine all possible ordered triples

(a,b,c)

that satisfy both equations. c. Part 3: (c) Verify algebraically that your solution set indeed yields continuity of

g

(-\infty,\infty)

. d. Part 4: (d) Explain why no other triples

(a,b,c)

can make

g

continuous at both

x=1

and

x=3

. Provide a logical argument rather than additional algebraic manipulation.