To solve for the inverse of the given function $f(x) = x^3 + 27$:

Step-by-step Solution:

1. Write the function as $y = f(x)$: $y = x^3 + 27$

2. Swap $x$ and $y$: To find the inverse, interchange $x$ and $y$: $x = y^3 + 27$

3. Solve for $y$: Subtract 27 from both sides: $x - 27 = y^3$

   Take the cube root of both sides: $y = \sqrt[3]{x - 27}$

   So, the inverse function is: $f^{-1}(x) = \sqrt[3]{x - 27}$

The placeholders in the question can now be filled in as: $f^{-1}(x) = \sqrt[3]{x - 27}$

• The "?" box represents the cube root ($\sqrt[3]{}$).
• The second box represents the term $-27$.

Let me know if you'd like me to explain any steps further!

Related Questions:

1. What is the general procedure for finding the inverse of a function?
2. How do we verify if two functions are inverses of each other?
3. What is the graph of $f(x) = x^3 + 27$, and how does it relate to its inverse?
4. What are the domain and range of $f^{-1}(x) = \sqrt[3]{x - 27}$?
5. How does the cube root function differ from a square root in terms of its domain?

Tip:

When finding the inverse of a function, always check whether the function is one-to-one (passes the horizontal line test) to ensure that an inverse exists.