Let F(X) =
5x-4, if x is less than or equal to 2
-4x+b, if x is greater than 2
If is a function which is continuous everywhere, then we must have b=_______?
intuitively, what does continuous mean? it means there are no gaps, which means, we want -4x + b to pick up exactly where 5x - 4 left off.
let $\displaystyle f_1(x) = 5x - 4$ and $\displaystyle f_2(x) = -4x + b$
we want, $\displaystyle f_1(2) = f_2(2)$
(yes, technically, $\displaystyle x = 2$ is not in the domain of $\displaystyle f_2(x)$, but we do this because we want to get arbitrary close to $\displaystyle x = 2$)