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Algebra Level 3

Let f ( x ) f(x) be a function such that f ( x ) = ∣ x 2 − 4 ∣ − 1 f(x)=|x^2-4|-1 and g ( x ) g(x) a function such that g ( x ) = f ( x ) g(x)=\sqrt{f(x)} . The domain of g ( x ) g(x) can be written as ( − ∞ , − a ] ∪ [ − b , c ] ∪ [ d , + ∞ ) (-\infty,-\sqrt{a}] \cup [-\sqrt{b},\sqrt{c}] \cup [\sqrt{d},+\infty) . Find the value of a + b + c + d a+b+c+d


The answer is 16.

Hjalmar Orellana Soto by Hjalmar Orellana Soto , 24, Chile

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1 solution

g ( x ) g(x) will be defined for f ( x ) = ∣ x 2 − 4 ∣ − 1 ≥ 0 f(x) = |x^2 - 4| - 1 \ge 0 and this happens if and only if ∣ x 2 − 4 ∣ ≥ 1 ⟺ x ∈ ( − ∞ , 5 ] ∪ [ − 3 , 3 ] ∪ [ 5 , ∞ ) |x^2 - 4| \ge 1 \iff x \in (-\infty, \sqrt{5}] \cup [- \sqrt{3}, \sqrt{3}] \cup [\sqrt{5},\infty) ∣ x 2 − 4 ∣ ≥ 1 ⟺ ( x 2 − 4 ≥ 1 or 4 − x 2 ≥ 1 ) ⟺ ( x 2 ≥ 5 or 3 ≥ x 2 ) |x^2 - 4| \ge 1 \iff (x^2 - 4 \ge 1 \text{ or } 4 - x^2 \ge 1) \iff (x^2 \ge 5 \text{ or } 3 \ge x^2)

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