Question

Find the constant a such that the function is continuous on the entire real line.

Answer 1

\[f(x) = 2, x \le -1\]\[f(x) = ax+b, -1<x<3\]
\[f(x) = -2, x \ge3\]

Answer 2

\[\text{Let}~~ f_1(x) = 2,~f_2(x) = ax+b,~ \text{and}~f_3(x) = -2\]For these to be continuous f1 = f2 at x=-1 and f2=f3 at x = 3. Therefore, \[f_2(-1) = 2 ~~\text{and} ~~ f_2(3) = -2 \]This can be expressed as \[2 = a(-1) + b ~~\text{and} ~~ -2 = a(3) + b\]Solving simultaneously for a and b, we find that a = -1 and b = 1

Answer 3

Hey, thanks! Don't know why I didn't see that before.