Question

Determine the values of a and b to make the following function continuous at every value of x.?

f(x)= { e^(x^2-x+a) if x < 1 , (4^x-x^2)) if 1<x<2 , 6In(x-b) if x>2

Answer 1

e^(x^2-x+a) if x < 1
as x->1-, f(x) -> e^a

(4^x-x^2)) if 1<x<2
as x->1+, f(x) -> 4^1-1 = 3
as x->2-, f(x) -> 4^2-2^2 = 12

6In(x-b) if x>2
as x->2-, f(x) -> 6ln(2-b)

so, we need

e^a = 3
a = ln3

6ln(2-b) = 12
ln(2-b)=2
2-b = e^2
b = 2-e^2

But using those values will not make f(x) continuous at every x, since f(x) is not defined at x=1 or x=2. To fix that, we need to add

f(1) = 3
f(2) = 12

Or make some of the intervals closed on one end or or maybe both ends.