(a)(i) We have the equations:

\(4 = a + \frac{1}{2}b\)

\(3 = a + b\)

Subtract the first equation from the second:

\(3 - 4 = a + b - (a + \frac{1}{2}b)\)

\(-1 = \frac{1}{2}b\)

\(b = -2\)

Substitute \(b = -2\) into \(3 = a + b\):

\(3 = a - 2\)

\(a = 5\)

(a)(ii) \(ff(x) = a + b \sin(a + b \sin x)\)

\(ff(0) = 5 - 2 \sin(5) = 6.92\)

(b) The range of \(g(x) = c + d \sin x\) is \(-4 \leq g(x) \leq 10\).

\(10 = c + d\) and \(-4 = c - d\)

Adding these equations:

\(10 - (-4) = c + d + c - d\)

\(14 = 2c\)

\(c = 7\)

Substitute \(c = 3\) into \(10 = c + d\):

\(10 = 3 + d\)

\(d = 7\)

Alternatively, \(d = -7\) also satisfies the range condition.