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Let f : R → R and g : R → R be defined as f(x) = { x + a, x < 0 { |x − 1|, x ≥ 0 and g(x) = { x + 1, x < 0 { (x − 1)^2 + b, x ≥ 0 where a, b are non-negative real numbers. If (g∘f)(x) is continuous for all x ∈ R, then a + b is equal to _____.

1. 0
2. 1
3. 2
4. 3

Correct answer: 0

Solution

For (g∘f)(x) to be continuous at x = 0, the outputs of f and g must match at that point. Since f(0) = |0 - 1| = 1 and g(1) = (1 - 1)^2 + b = b, setting b = 0 ensures continuity, and since a must also be 0 to maintain continuity for x < 0, we find that a + b = 0.

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