(i) The function f(x) = 4x − 2x² is a quadratic function. The vertex form of a quadratic function ax² + bx + c is given by completing the square or using calculus to find the turning point. The turning point occurs at x = 1, and the maximum value of f(x) is 2. Therefore, the range of f is ≤ 2.

(ii) To find the value of k for which gf(x) = k has equal roots, substitute f(x) into g(x):

gf(x) = g(f(x)) = 5(4x − 2x²) + 3 = 20x − 10x² + 3.

Set this equal to k:

\(−10x² + 20x + 3 = k.\)

For equal roots, the discriminant must be zero:

\(b² − 4ac = 0, where a = −10, b = 20, c = 3 − k.\)

\(20² − 4(−10)(3 − k) = 0.\)

\(400 + 40(3 − k) = 0.\)

\(400 + 120 − 40k = 0.\)

\(520 = 40k.\)

\(k = 13.\)