To find the inverse of each given function $F(x)$, we follow these general steps:

1. Replace $F(x)$ with $y$.
2. Swap $x$ and $y$.
3. Solve for $y$.

Let's determine the inverse of each function:

1. $F(x) = (2x - 6)^2 + 9$

Step-by-step:

1. Replace $F(x)$ with $y$: $y = (2x - 6)^2 + 9$
2. Swap $x$ and $y$: $x = (2y - 6)^2 + 9$
3. Solve for $y$: $x - 9 = (2y - 6)^2$ $\sqrt{x - 9} = 2y - 6$ $y = \frac{\sqrt{x - 9} + 6}{2} \quad \text{and} \quad y = \frac{-\sqrt{x - 9} + 6}{2}$ Since squaring introduces both positive and negative roots, the inverse is a piecewise function.

Inverse: $F^{-1}(x) = \frac{\sqrt{x - 9} + 6}{2}$ and $F^{-1}(x) = \frac{-\sqrt{x - 9} + 6}{2}$

---

2. $F(x) = (6x + 12)^2$

Step-by-step:

1. Replace $F(x)$ with $y$: $y = (6x + 12)^2$
2. Swap $x$ and $y$: $x = (6y + 12)^2$
3. Solve for $y$: $\sqrt{x} = 6y + 12$ $y = \frac{\sqrt{x} - 12}{6} \quad \text{and} \quad y = \frac{-\sqrt{x} - 12}{6}$

Inverse: $F^{-1}(x) = \frac{\sqrt{x} - 12}{6}$ and $F^{-1}(x) = \frac{-\sqrt{x} - 12}{6}$

---

3. $F(x) = \frac{2x - 3}{6x - 1}$

Step-by-step:

1. Replace $F(x)$ with $y$: $y = \frac{2x - 3}{6x - 1}$
2. Swap $x$ and $y$: $x = \frac{2y - 3}{6y - 1}$
3. Solve for $y$: $x(6y - 1) = 2y - 3$ $6xy - x = 2y - 3$ $6xy - 2y = x - 3$ $y(6x - 2) = x - 3$ $y = \frac{x - 3}{6x - 2}$

Inverse: $F^{-1}(x) = \frac{x - 3}{6x - 2}$

---

4. $F(x) = \frac{x}{3 + 7x}$

Step-by-step:

1. Replace $F(x)$ with $y$: $y = \frac{x}{3 + 7x}$
2. Swap $x$ and $y$: $x = \frac{y}{3 + 7y}$
3. Solve for $y$: $x(3 + 7y) = y$ $3x + 7xy = y$ $7xy - y = -3x$ $y(7x - 1) = -3x$ $y = \frac{-3x}{7x - 1}$

Inverse: $F^{-1}(x) = \frac{-3x}{7x - 1}$

---

5. $F(x) = \frac{3x + 6}{2x - 9}$

Step-by-step:

1. Replace $F(x)$ with $y$: $y = \frac{3x + 6}{2x - 9}$
2. Swap $x$ and $y$: $x = \frac{3y + 6}{2y - 9}$
3. Solve for $y$: $x(2y - 9) = 3y + 6$ $2xy - 9x = 3y + 6$ $2xy - 3y = 9x + 6$ $y(2x - 3) = 9x + 6$ $y = \frac{9x + 6}{2x - 3}$

Inverse: $F^{-1}(x) = \frac{9x + 6}{2x - 3}$

---

Would you like more details on any of these solutions or have any questions?

Here are 5 related questions to expand on the concept:

1. How do we determine the domain and range of an inverse function?
2. What happens to the graph of a function when you find its inverse?
3. Are there functions that do not have inverses? If so, why?
4. How can we verify that a function is the inverse of another?
5. Why do some functions produce multiple inverses when squared?

Tip: When finding inverses of functions, always check that the function is one-to-one (passes the horizontal line test), as this guarantees the existence of an inverse.