Question Detail

For f(x) to be continuous, the function must meet at the points where its definition changes (x=1 and x=2).1. Continuity at x=1: The left-hand limit, right-hand limit, and function value must be equal. lim (x→1⁻) f(x) = lim (x→1⁻) (x+1) = 1+1 = 2 lim (x→1⁺) f(x) = lim (x→1⁺) (ax+b) = a(1)+b = a+b f(1) = a(1)+b = a+b Therefore, a + b = 2 (Equation 1)2. Continuity at x=2: The left-hand limit, right-hand limit, and function value must be equal. lim (x→2⁻) f(x) = lim (x→2⁻) (ax+b) = a(2)+b = 2a+b lim (x→2⁺) f(x) = lim (x→2⁺) (3x) = 3(2) = 6 f(2) = 3(2) = 6 Therefore, 2a + b = 6 (Equation 2)3. Solve the system of equations: Subtract Equation 1 from Equation 2: (2a + b) - (a + b) = 6 - 2 a = 4 Substitute a = 4 into Equation 1: 4 + b = 2 b = 2 - 4 b = -2So, the values are a=4 and b=-2.