Question

(a) let f : R → R be a function defined by
f(x) = x + 4 if x ≤ 1
ax + b if 1 < x ≤ 3
3x − 8 if x > 3
Find the values of a and b that makes f(x) continuous on R.

Answer 1

The function should be continuous, so $\lim_{x\to 1+} (ax+b) = f(1) \;\; \mathrm{and} \;\; \lim_{x\to 3+} (3x-8) = f(3).$ Therefore, $a+b = 1+4, \;\; 9-8 = 3a+b.$

$a+b=5, \;\; 1=3a+b.$ Solving this system, we get , $a = -2$ and $b=7.$