Mathematics

Problem

Define a function

g

whose graph consists of three segments described below. • Segment 1: the straight line joining

(-3,1)

(-1,3)

, including the endpoint at

x=-3

but excluding the point

x=-1

. • Segment 2: the horizontal line

y=3

for

-1\le x\le 2

. • Segment 3: the curve

y=(x-2)^2+1

for

x>2

, with the point

(2,1)

excluded. Use this description to answer the following questions. a. (i) Sketch the graph of

g

in the interval

-3\le x\le 4

, clearly indicating all open and closed points. b. (ii) Evaluate

\displaystyle \lim_{x\to (-1)^-}g(x)

and

\displaystyle \lim_{x\to (-1)^+}g(x)

. c. (iii) Evaluate

\displaystyle \lim_{x\to 2^-}g(x)

and

\displaystyle \lim_{x\to 2^+}g(x)

. d. (iv) For each of the points

x=-1

and

x=2

, state whether

g

is continuous, giving reasons based on your results from parts (ii) and (iii).