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MTH 102 MTH 102 Past Questions Objective Question
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Question

Find the values of a and b that make the function f(x) continuous, where:
f(x) = x+1 for x < 1
f(x) = ax+b for 1 ≤ x < 2
f(x) = 3x for x ≥ 2
Options
A <code>a = -2, b = 4</code>
B <code>a = 4, b = -2</code>
Correct Answer
C <code>a = 2, b = 4</code>
D <code>a = -2, b = -4</code>
Correct Answer

Option B is the correct answer.

Detailed Explanation

For f(x) to be continuous, the pieces of the function must meet at the boundaries.
  1. Continuity at x = 1:The left-hand limit, the function value, and the right-hand limit must be equal.lim_{x→1⁻} f(x) = lim_{x→1⁻} (x+1) = 1+1 = 2f(1) = a(1)+b = a+bSo, a+b = 2 (Equation 1)
  2. Continuity at x = 2:The left-hand limit, the function value, and the right-hand limit must be equal.lim_{x→2⁻} f(x) = lim_{x→2⁻} (ax+b) = a(2)+b = 2a+bf(2) = 3(2) = 6So, 2a+b = 6 (Equation 2)
  3. Solve the system of linear equations: Equation 1: a + b = 2 Equation 2: 2a + b = 6 Subtract Equation 1 from Equation 2:(2a + b) - (a + b) = 6 - 2a = 4 Substitute a = 4 into Equation 1: 4 + b = 2b = 2 - 4b = -2
Thus, the values that make the function continuous are a = 4 and b = -2.

Hint

For a piecewise function to be continuous, the function values must match at the transition points. Set up equations for continuity at x=1 and x=2 by equating the function's definition from each side.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 439 of 930
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