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Find the domain of each function: (i) y = 3/(4 - x²) + log10(x³ - x) (ii) y = 1/sqrt(sin x) + cube_root(sin x)

1. (i) (-1,0) U (1,2) U (2, infinity); (ii) x in (2n*pi, (2n+1)*pi), n integer
2. (i) (-1,1); (ii) all real x
3. (i) (0, infinity); (ii) x in [2n*pi, (2n+1)*pi]
4. (i) (-2,2); (ii) x in ((2n-1)*pi, 2n*pi)

Correct answer: (i) (-1,0) U (1,2) U (2, infinity); (ii) x in (2n*pi, (2n+1)*pi), n integer

Solution

For (i), the logarithm requires its argument positive and the rational term requires the denominator nonzero. Solving x(x-1)(x+1) > 0 gives (-1,0) U (1, infinity), and removing x = 2 (where 4 - x² = 0; x = -2 already excluded) gives the domain. For (ii), 1/sqrt(sin x) requires sin x > 0, and the cube root is defined everywhere, so the domain is where sin x > 0.

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