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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!

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### 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 \( y = x \).

The function f(x) = ³√(x + 3) - 8 is one-to-one and therefore invertible. Determine f⁻¹. Graph both f and f⁻¹, along with the identity function f(x) = x.

Learn how to find the inverse function of f(x) = ³√(x + 3) - 8, graph the original and inverse functions, and understand the importance of the identity function f(x) = x. This problem covers key algebraic concepts, including inverse functions and their graphical relationship.

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