The problem you have uploaded contains four parts asking to find the inverse of given functions:

Problem 11: Find $f^{-1}(x)$ (the inverse of the function).

The parts are as follows:

(a) $f(x) = \sqrt{x} + 4$

(b) $f(x) = \frac{1 - x}{2 + x}$

(c) $f(x) = (x - 4)^7$

(d) A graph is shown for $f(x)$, and you likely need to determine $f^{-1}(x)$ using the graph.

Let's go through the inverses step by step:

(a) $f(x) = \sqrt{x} + 4$

To find the inverse:

1. Start by replacing $f(x)$ with $y$:
   $y = \sqrt{x} + 4$

2. Solve for $x$ in terms of $y$: $y - 4 = \sqrt{x}$ Squaring both sides: $(y - 4)^2 = x$

3. Now, swap $x$ and $y$ to express the inverse function: $f^{-1}(x) = (x - 4)^2$

(b) $f(x) = \frac{1 - x}{2 + x}$

For the inverse:

1. Replace $f(x)$ with $y$: $y = \frac{1 - x}{2 + x}$

2. Multiply both sides by $2 + x$ to eliminate the denominator: $y(2 + x) = 1 - x$

3. Expand and solve for $x$: $2y + xy = 1 - x$ $xy + x = 1 - 2y$ Factor out $x$: $x(y + 1) = 1 - 2y$ $x = \frac{1 - 2y}{y + 1}$

4. Swap $x$ and $y$: $f^{-1}(x) = \frac{1 - 2x}{x + 1}$

(c) $f(x) = (x - 4)^7$

This function involves a power. To find the inverse:

1. Replace $f(x)$ with $y$: $y = (x - 4)^7$

2. Take the seventh root of both sides: $\sqrt[7]{y} = x - 4$

3. Solve for $x$: $x = \sqrt[7]{y} + 4$

4. Swap $x$ and $y$: $f^{-1}(x) = \sqrt[7]{x} + 4$

(d) Graph

For part (d), you'll need to analyze the graph. The inverse of a function is a reflection of the original function's graph across the line $y = x$. If you describe the shape of the graph, I can help you further interpret it.

Would you like more details on any of these solutions?

Here are 5 related questions to deepen your understanding:

1. How do you verify if a function has an inverse?
2. What are the conditions for a function to have an inverse?
3. How does the graphical method of finding an inverse work?
4. Can you find the domain and range of each inverse function?
5. Why does squaring both sides when solving for the inverse introduce restrictions?

Tip: Always check that the inverse function satisfies the original function by verifying $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.