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Let f(g(x)) be defined piecewise as: f(g(x)) = -(2x + 1) for -2 <= x <= -1/2, f(g(x)) = 2x + 1 for -1/2 <= x <= 3, and f(g(x)) = 5x - 5 for 3 < x <= 9/2. The range of f(g(x)) on [-2, 9/2] is [0, 13]. For how many values of lambda in the interval (1, 34] does the equation f(|x|) + g(|x|) = lambda have exactly two distinct real solutions?

1. (A) 32
2. (B) 33
3. (C) 34
4. (D) 31

Correct answer: (B) 33

Solution

Since |x| >= 0, only the pieces for x >= 0 matter: the combined function h(x) = f(x) + g(x) rises from h(0) = 1, climbs linearly. Reflecting about x = 0 means h(|x|) is even. For exactly two real solutions, the line lambda = c must intersect the right-half graph at exactly one positive x value (the symmetric left side gives the second). Excluding lambda = 1 (vertex, four solutions) and the endpoint, there are 33 integer values in (1, 34].

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