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Determine the domain of definition of each function: (i) y = sqrt(-p x) with p > 0; (ii) y = sqrt(1 - |x|); (iii) y = 1/(x² + 1); (iv) y = 1/(x³ - x); (v) y = sqrt(x² - 4x + 3); (vi) y = x/sqrt(x² - 3x + 2).

1. (i) x <= 0; (ii) -1 <= x <= 1; (iii) all real x; (iv) x != 0, 1, -1; (v) x <= 1 or x >= 3; (vi) x < 1 or x > 2
2. (i) x >= 0; (ii) x <= 1; (iii) x != 1; (iv) x != 0; (v) 1 <= x <= 3; (vi) 1 < x < 2
3. (i) all real x; (ii) -1 < x < 1; (iii) x > 0; (iv) x != 1, -1; (v) all real x; (vi) x <= 1 or x >= 2
4. (i) x <= 0; (ii) -1 <= x <= 1; (iii) x != 0; (iv) x != 0, 1, -1; (v) 1 <= x <= 3; (vi) 1 <= x <= 2

Correct answer: (i) x <= 0; (ii) -1 <= x <= 1; (iii) all real x; (iv) x != 0, 1, -1; (v) x <= 1 or x >= 3; (vi) x < 1 or x > 2

Solution

Each part applies the basic domain rules. (i) -p x >= 0 with p > 0 means x <= 0. (ii) 1 - |x| >= 0 means |x| <= 1. (iii) denominator x²+1 never zero, so all reals. (iv) x(x-1)(x+1) != 0. (v) (x-1)(x-3) >= 0 means x <= 1 or x >= 3. (vi) (x-1)(x-2) > 0 (strict, in denominator under root) means x < 1 or x > 2.

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