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Find the domain of the function y = sqrt((e^x - 1)*(x² - 4)).

1. [-2, 0] U [2, infinity)
2. (-infinity, -2] U [0, 2]
3. [-2, 2]
4. [0, infinity)

Correct answer: [-2, 0] U [2, infinity)

Solution

We need (e^x - 1)(x² - 4) >= 0. The factor e^x - 1 is >=0 for x>=0 and <0 for x<0. The factor x² - 4 is >=0 for |x|>=2 and <0 for -2<x<2. For x>=0 the product is non-negative only when x²-4>=0, i.e. x>=2 (x=0 gives 0, included). For x<0, e^x-1<0, so we need x²-4<=0, i.e. -2<=x<0. Combining: [-2,0] U [2, infinity).

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