The problem you're working on involves determining the inverse of the function $f(x) = \sqrt[3]{x + 3} - 8$, graphing both $f(x)$ and its inverse $f^{-1}(x)$, and sketching the identity function $f(x) = x$ on the same graph.

Let's break this down step by step:

Step 1: Finding the Inverse Function

Given $f(x) = \sqrt[3]{x + 3} - 8$, we need to find $f^{-1}(x)$.

1. Start by replacing $f(x)$ with $y$: $y = \sqrt[3]{x + 3} - 8$
2. Solve for $x$ in terms of $y$: $y + 8 = \sqrt[3]{x + 3}$
3. Cube both sides to eliminate the cube root: $(y + 8)^3 = x + 3$
4. Solve for $x$: $x = (y + 8)^3 - 3$

Thus, the inverse function is: $f^{-1}(x) = (x + 8)^3 - 3$

Step 2: Graphing the Functions

Now that we have the inverse function $f^{-1}(x) = (x + 8)^3 - 3$, you are asked to plot the following:

1. The original function $f(x) = \sqrt[3]{x + 3} - 8$.
2. The inverse function $f^{-1}(x) = (x + 8)^3 - 3$.
3. The identity function $f(x) = x$, which is simply a diagonal line passing through the origin.

Step 3: Characteristics for Graphing

1. $f(x) = \sqrt[3]{x + 3} - 8$ will have a horizontal shift of 3 units to the left and 8 units down from the basic cube root function.
2. $f^{-1}(x) = (x + 8)^3 - 3$ will look like a cubic curve, shifted 8 units to the left and 3 units down.
3. The identity function $f(x) = x$ is a straight line passing through points like (0, 0), (1, 1), (-1, -1), etc.

If you'd like, I can help you further in plotting these functions or provide more guidance. Let me know if you'd like a detailed visual explanation or further help with graphing!

Relative Questions:

1. What are the key properties of an inverse function?
2. How does the graph of a function relate to its inverse geometrically?
3. What are the transformations applied to $f(x) = \sqrt[3]{x}$ to get the function $f(x) = \sqrt[3]{x + 3} - 8$?
4. Why is the identity function $f(x) = x$ important when comparing a function and its inverse?
5. How can you verify that two functions are inverses of each other algebraically?

Tip:

When graphing a function and its inverse, remember that the two graphs will always be symmetric with respect to the line $y = x$.