The image asks to find the inverse of the following functions:

a. $F(x) = \sqrt{x + 6}$

To find the inverse, follow these steps:

1. Replace $F(x)$ with $y$: $y = \sqrt{x + 6}$
2. Swap $x$ and $y$: $x = \sqrt{y + 6}$
3. Square both sides: $x^2 = y + 6$
4. Solve for $y$: $y = x^2 - 6$

Thus, the inverse is: $F^{-1}(x) = x^2 - 6$

b. $F(x) = \sqrt[3]{2x - 6}$

1. Replace $F(x)$ with $y$: $y = \sqrt[3]{2x - 6}$
2. Swap $x$ and $y$: $x = \sqrt[3]{2y - 6}$
3. Cube both sides: $x^3 = 2y - 6$
4. Solve for $y$: $2y = x^3 + 6$ $y = \frac{x^3 + 6}{2}$

Thus, the inverse is: $F^{-1}(x) = \frac{x^3 + 6}{2}$

c. $F(x) = \frac{-4x + 2}{2x + 5}$

1. Replace $F(x)$ with $y$: $y = \frac{-4x + 2}{2x + 5}$
2. Swap $x$ and $y$: $x = \frac{-4y + 2}{2y + 5}$
3. Cross-multiply: $x(2y + 5) = -4y + 2$
4. Expand both sides: $2xy + 5x = -4y + 2$
5. Collect all terms involving $y$ on one side: $2xy + 4y = 2 - 5x$ $y(2x + 4) = 2 - 5x$
6. Solve for $y$: $y = \frac{2 - 5x}{2x + 4}$

Thus, the inverse is: $F^{-1}(x) = \frac{2 - 5x}{2x + 4}$

d. $F(x) = \frac{3}{5} (2x + 3)$

1. Replace $F(x)$ with $y$: $y = \frac{3}{5}(2x + 3)$
2. Multiply both sides by 5: $5y = 3(2x + 3)$
3. Expand: $5y = 6x + 9$
4. Solve for $x$: $6x = 5y - 9$ $x = \frac{5y - 9}{6}$

Thus, the inverse is: $F^{-1}(x) = \frac{5x - 9}{6}$

Would you like more details or explanations on any of the steps?

Here are 5 related questions:

1. What is the general process for finding the inverse of a function?
2. How do we know if a function has an inverse?
3. What are the properties of an inverse function?
4. How do you check if your inverse function is correct?
5. Can all functions be inverted? Why or why not?

Tip: When finding an inverse function, swapping $x$ and $y$ is the key first step in solving for the inverse.