The relationship between a and b for the continuous function f(x)= array ll a x+1, & if x 3 b x+3, & if x>3 array . at x=3 is

Step-by-step solution

For the function f(x) to be continuous at x = 3 , the left-hand limit, the right-hand limit, and the value of the function at x = 3 must be equal. The left-hand limit at x = 3 is given by _ x 3⁻ f(x) = _ x 3⁻ (ax + 1) = 3a + 1 . The value of the function at x = 3 is f(3) = a(3) + 1 = 3a + 1 . The right-hand limit at x = 3 is given by _ x 3⁺ f(x) = _ x 3⁺ (bx + 3) = 3b + 3 . Equating the left-hand limit and the right-hand limit for continuity: 3a + 1 = 3b + 3 3a = 3b + 2 . Answer: 3 a=3 b+2