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Find the domain of each function: (i) y = 1 - log10(x); (ii) y = 1/sqrt(x² - 4x).

1. (i) x > 0; (ii) x < 0 or x > 4
2. (i) x >= 0; (ii) 0 < x < 4
3. (i) x > 0; (ii) 0 <= x <= 4
4. (i) x != 0; (ii) x < 0 or x > 4

Correct answer: (i) x > 0; (ii) x < 0 or x > 4

Solution

(i) log10(x) is defined only for x > 0, so the domain is (0, infinity). (ii) The expression under the root, x² - 4x = x(x-4), must be > 0 (strict, since it sits in a denominator). x(x-4) > 0 holds when x < 0 or x > 4. Domain: (-infinity, 0) union (4, infinity).

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