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Find the domain and range of each function: (i) f(x) = x/(1 + |x|) (ii) y = sqrt(2 - x) + sqrt(1 + x) (iii) f(x) = (sqrt(x + 4) - 3)/(x - 5)

1. (i) domain R, range (-1,1); (ii) domain [-1,2], range [sqrt(3), sqrt(6)]; (iii) domain [-4,5) U (5, infinity), range (0, 1/3] minus {1/6}
2. (i) domain R, range [-1,1]; (ii) domain (-1,2), range (sqrt(3), sqrt(6)); (iii) domain [-4, infinity), range R
3. (i) domain R minus {0}, range (-1,1); (ii) domain [-1,2], range [0, sqrt(6)]; (iii) domain (5, infinity), range R
4. (i) domain (-1,1), range R; (ii) domain [2, infinity), range [sqrt(3), infinity); (iii) domain [-4,5), range R minus {1/6}

Correct answer: (i) domain R, range (-1,1); (ii) domain [-1,2], range [sqrt(3), sqrt(6)]; (iii) domain [-4,5) U (5, infinity), range (0, 1/3] minus {1/6}

Solution

Standard domain-range analysis: restrict by non-negativity of radicands and non-zero denominators, then determine attainable output values, watching for excluded limiting values.

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