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Find the range of each function: (i) f(x) = (x - 1)/(x + 2); (ii) f(x) = sin² x - 2 sin x + 4; (iii) f(x) = sin(log₂ x); (iv) f(x) = (e^(2x) - e^x + 1)/(e^(2x) + e^x + 1). Which option lists the four ranges correctly?

1. (i) R - {1} (ii) [3, 7] (iii) [-1, 1] (iv) (3/5, 1)
2. (i) R - {1} (ii) [3, 7] (iii) (-1, 1) (iv) [3/5, 1)
3. (i) R (ii) [3, 7] (iii) [-1, 1] (iv) (3/5, 1)
4. (i) R - {1} (ii) [2, 6] (iii) [-1, 1] (iv) (1/3, 3)

Correct answer: (i) R - {1} (ii) [3, 7] (iii) [-1, 1] (iv) (3/5, 1)

Solution

Each range follows from a standard substitution. The rational function in (i) takes all values except the horizontal asymptote; the trig quadratic in (ii) is evaluated over t in [-1,1]; (iii) since log₂ x spans all reals, sine takes its full range; (iv) reduces to a bounded expression in t = e^x > 0.

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