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Find the range of each function: (i) f(x) = log base 4 of |x + 1/x|; (ii) f(x) = sin(3x² + 1); (iii) f(x) = 2 sin(2x + pi/4); (iv) f(x) = cos(2x + pi/4).

1. (i) [1/2, inf); (ii) [-1, 1]; (iii) [-2, 2]; (iv) [-1, 1]
2. (i) (0, inf); (ii) [0, 1]; (iii) [-1, 1]; (iv) [0, 1]
3. (i) [1, inf); (ii) [-1, 1]; (iii) [-2, 2]; (iv) [-1, 1]
4. (i) [1/2, inf); (ii) [sin 1, 1]; (iii) [-2, 2]; (iv) [-1, 1]

Correct answer: (i) [1/2, inf); (ii) [-1, 1]; (iii) [-2, 2]; (iv) [-1, 1]

Solution

(i) For real x != 0, |x + 1/x| >= 2, so log₄ of it is >= log₄(2) = 1/2, giving [1/2, inf). (ii) As x varies, 3x² + 1 takes all values >= 1, an interval of length > 2 pi, so sin covers the full [-1, 1]. (iii) Amplitude 2 gives [-2, 2]. (iv) cos gives [-1, 1].

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