To help you with this, I'll break down each part of the question as follows:

(a) Part i: Domain and Range of $f(x) = 3x - 2 + 5$

1. Simplify the expression for $f(x)$: $f(x) = 3x - 2 + 5 = 3x + 3$ So, the function simplifies to $f(x) = 3x + 3$.

2. Domain of $f(x)$: Since $f(x) = 3x + 3$ is a linear function, there are no restrictions on $x$. Thus, the domain of $f(x)$ is: $\text{Domain of } f(x): \quad x \in \mathbb{R}$

3. Range of $f(x)$: Similarly, because $f(x) = 3x + 3$ is a linear function, it can take all real values as $x$ varies over all real numbers. Therefore, the range of $f(x)$ is also: $\text{Range of } f(x): \quad f(x) \in \mathbb{R}$

(a) Part ii: Relationship Between $f$ and $f^{-1}$

The inverse function $f^{-1}$ is found by swapping $x$ and $y$ in the equation and solving for $y$:

1. Start with $y = 3x + 3$.

2. Swap $x$ and $y$ to get $x = 3y + 3$.

3. Solve for $y$: $x - 3 = 3y \Rightarrow y = \frac{x - 3}{3}$ So, the inverse function $f^{-1}(x)$ is: $f^{-1}(x) = \frac{x - 3}{3}$

4. Graphing the Relationship: The graph of $f(x) = 3x + 3$ and $f^{-1}(x) = \frac{x - 3}{3}$ will be reflections of each other over the line $y = x$. This line acts as the mirror line between a function and its inverse.

(b) Given Functions $g(x) = x + 1$ and $h(x) = ax + 3$

You mentioned functions $g(x) = x + 1$ and $h(x) = ax + 3$. Please provide any additional details or questions regarding how these functions are to be used—whether they need composition, comparison, or specific evaluations—so I can continue with the solution.

Let me know if you need a graphical representation or have questions on any part! Here are some further questions to deepen understanding:

1. How do you find the inverse of a function, and why does swapping $x$ and $y$ help?
2. What general features of linear functions affect their domain and range?
3. What other types of functions have restricted domains and ranges?
4. How can you confirm if two functions are inverses by looking at their graphs?
5. What are other ways to show the relationship between a function and its inverse?

Tip: For any function $f(x)$, finding the inverse function involves reversing the operations applied to $x$, ensuring that the resulting function undoes what $f(x)$ did.