How do i find the constants a and b so the function is continuous on the entire real number line

f(x)= {2 x<=-1
.........{ax+b -1 .........{-2 x>=3

Evaluate the function at the points x=-1, and x=3, and enforce continuity by setting the two expressions equal.

at x=-1,

2 = a*(-1) + b
==>
b-a = 2

at x = 3,

a*3 + b = -2
==>
3a+b = -2

solving the system for a and b gives

b-a = 2
3a+b = -2
==>

a=-1
b= 1

Thus,

f(x)= {2 x<=-1
.........{-x+1 -1 .........{-2 x>=3