Mathematics

Problem

Define the piecewise function

h(x)=\begin{cases} ax+b, & x<1,\\ x^2-2, & 1\le x<3,\\ \sqrt{x+1}, & x\ge3, \end{cases}

where

a

and

b

are real constants. a. Part 1: (a) Determine all ordered pairs

(a,b)

such that

h

is continuous for all real numbers

x

. Show all work leading to your pair(s). b. Part 2: (b) For the values of

a

and

b

found in part (a), evaluate

h(1)

and

h(3)

, and explain how these values confirm continuity at the transition points.