CA Foundation · Quantitative Aptitude · Sets, Relations and Functions, Limits and Continuity
A function is defined by f(x) = ax + b for x ≤ 1, f(x) = 7 for 1 < x < 3, and f(x) = 2ax − b for x ≥ 3. If f is continuous everywhere, what is the value of a + b?
The value of a + b is 7. Continuity at x = 1 gives a + b = 7, and continuity at x = 3 gives 6a − b = 7. Solving gives a = 2 and b = 5, so a + b = 7, which also satisfies both equations.
- A7Correct
- B6
- C5
- D8
Explanation
Continuity at x = 1: a + b = 7. Continuity at x = 3: 2a(3) − b = 7, so 6a − b = 7. Adding the two equations gives 7a = 14, so a = 2, and b = 5. Then a + b = 7. Check: 6(2) − 5 = 7, which holds.
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