Question

F(x) = 2x+5 if x<=1, -2x+1 if x>=-1 is continuous at x = -1. t or f

Answer 1

The given piecewise-defined function f(x),

`f(x) = \begin{cases} 2x + 5 & \text{if } x \le -1 \\ -2x + 1 & \text{if } x \ge -1\end{cases}`

is continuous at x = -1 if both limits as x approaches -1 from either side are the same. We have

`\displaystyle \lim_(x\to-1^-) f(x) = \lim_(x\to-1) (2x+5) = 3`

`\displaystyle \lim_(x\to-1^+) f(x) = \lim_(x\to-1) (-2x+1) = 3`

so the claim is true, f(x) is indeed continuous at x = -1.