Let's solve these problems step by step.

1) Given that $f(x)$ is a linear function where $f(2) = 9$ and $f(-2) = -11$, determine:

(i) $f(x)$

Since $f(x)$ is linear, it can be written as: $f(x) = mx + c$ where $m$ is the slope and $c$ is the y-intercept.

1. Find the slope ($m$): Using the formula for slope between two points: $m = \frac{f(2) - f(-2)}{2 - (-2)} = \frac{9 - (-11)}{4} = \frac{20}{4} = 5$

2. Substitute $m = 5$ into $f(x) = mx + c$: $f(x) = 5x + c$

3. Find $c$ using one of the points, say $(2, 9)$: Substitute $x = 2$ and $f(x) = 9$: $9 = 5(2) + c \implies 9 = 10 + c \implies c = -1$

4. Final equation: $f(x) = 5x - 1$

(ii) $f(0.2)$

Substitute $x = 0.2$ into $f(x) = 5x - 1$: $f(0.2) = 5(0.2) - 1 = 1 - 1 = 0$

2) Given that $f(x)$ is a linear function where $f(2) + f(3) = 0$ and $f(5) = 5$, determine:

(i) $f(x)$

1. Express $f(x)$: Again, let: $f(x) = mx + c$

2. Use the condition $f(2) + f(3) = 0$: Substitute $f(2) = 2m + c$ and $f(3) = 3m + c$: $(2m + c) + (3m + c) = 0 \implies 5m + 2c = 0 \implies 2c = -5m \implies c = -\frac{5m}{2}$

3. **Use the condition ( f(5) =