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Find the range of each of the following functions: (i) f(x) = e^((x-1)²) (ii) f(x) = x³ - x² + x + 1 (iii) f(x) = log(x⁸ + x⁴ + x² + 1).

1. (i) [1, infinity) (ii) R (iii) [0, infinity)
2. (i) (0, infinity) (ii) R (iii) R
3. (i) [1, infinity) (ii) (0, infinity) (iii) [0, infinity)
4. (i) [e, infinity) (ii) R (iii) (0, infinity)

Correct answer: (i) [1, infinity) (ii) R (iii) [0, infinity)

Solution

(i) The exponent ranges over [0, infinity), so e raised to it ranges over [1, infinity). (ii) The cubic's derivative is always positive (discriminant 4 - 12 < 0), so it is strictly increasing with range all of R. (iii) The polynomial x⁸ + x⁴ + x² + 1 has minimum value 1 at x = 0 and increases to infinity, so its log ranges over [0, infinity).

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