Question Detail
Question
Find the values of a and b that make f(x) continuous, where f(x) is defined as:
f(x) = x+1, for x < 1
f(x) = ax+b, for 1 ≤ x < 2
f(x) = 3x, for x ≥ 2
f(x) = x+1, for x < 1
f(x) = ax+b, for 1 ≤ x < 2
f(x) = 3x, for x ≥ 2
Correct Answer
Option B is the correct answer.
Detailed Explanation
For a piecewise function to be continuous, the function must be continuous at its transition points. Here, the transition points are x=1 and x=2.For continuity at x=1:The left-hand limit must equal the function value at x=1.limx→1- f(x) = limx→1- (x+1) = 1+1 = 2.f(1) = a(1)+b = a+b.So, for continuity at x=1, we must have a+b = 2. (Equation 1)For continuity at x=2:The left-hand limit must equal the function value at x=2.limx→2- f(x) = limx→2- (ax+b) = a(2)+b = 2a+b.f(2) = 3(2) = 6.So, for continuity at x=2, we must have 2a+b = 6. (Equation 2)Now we have a system of two linear equations:1) a + b = 22) 2a + b = 6Subtract Equation 1 from Equation 2:(2a + b) - (a + b) = 6 - 2a = 4Substitute the value of a=4 into Equation 1:4 + b = 2b = 2 - 4b = -2Thus, the values that make the function continuous are a=4 and b=-2. This matches option B.
Hint
For a piecewise function to be continuous, the function values and limits must match at each transition point. Set up equations for continuity at x=1 and x=2, then solve the system of equations for 'a' and 'b'.