To tackle this problem, let's go through each part systematically. Based on your description, I'll make some assumptions about missing details and provide solutions accordingly.

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Given Function $f(x) = 3x - 2 + 5$

We start by simplifying this expression: $f(x) = 3x - 2 + 5 = 3x + 3$

i) Domain and Range of $f(x) = 3x + 3$

1. Domain: Since $f(x) = 3x + 3$ is a linear function, it is defined for all real values of $x$. Therefore: $\text{Domain of } f = (-\infty, \infty)$

2. Range: Since $f(x) = 3x + 3$ is also a linear function without any restrictions, it can output any real number as $x$ varies over all real numbers. Thus: $\text{Range of } f = (-\infty, \infty)$

ii) Relationship Between $f$ and $f^{-1}$

To find the inverse function $f^{-1}$ of $f(x) = 3x + 3$, follow these steps:

1. Replace $f(x)$ with $y$: $y = 3x + 3$

2. Solve for $x$ in terms of $y$: $y - 3 = 3x$ $x = \frac{y - 3}{3}$

3. Replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = \frac{x - 3}{3}$

Now, $f(x) = 3x + 3$ and $f^{-1}(x) = \frac{x - 3}{3}$.

To represent the relationship between $f$ and $f^{-1}$ graphically, note that their graphs are reflections of each other across the line $y = x$. Plotting both functions on the same coordinate plane would show this reflective property.

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

For this part, we’ll need additional information about the variable $a$ or further questions to proceed meaningfully with $g$ and $h$.

Would you like to continue with assumptions about $a$ or provide additional information for part (b)? Let me know if you’d like a graph as well.

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Relative Questions

1. What is the graphical significance of the line $y = x$ in relation to a function and its inverse?
2. How do the slopes of $f(x)$ and $f^{-1}(x)$ relate to each other?
3. Why is the domain and range of a linear function like $3x + 3$ unrestricted?
4. How does the concept of reflection across $y = x$ apply to inverse functions in general?
5. What changes if $f(x)$ is a non-linear function in terms of finding its inverse?

Tip

Remember, the inverse function $f^{-1}$ exists only if $f$ is bijective (both injective and surjective). This requirement is met if $f$ is strictly monotonic.