The given function is $f(x) = (x + 3)^5$, and we need to find the inverse function $f^{-1}(x)$.

Steps to Find the Inverse:

1. Replace $f(x)$ with $y$: $y = (x + 3)^5$

2. Swap $x$ and $y$ to find the inverse relation: $x = (y + 3)^5$

3. Solve for $y$:

   • Take the fifth root of both sides to remove the exponent: $\sqrt[5]{x} = y + 3$
   • Isolate $y$: $y = \sqrt[5]{x} - 3$

4. Replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = \sqrt[5]{x} - 3$

Final Answer:

$f^{-1}(x) = \sqrt[5]{x} - 3$

Let me know if you'd like more details or explanations!

Related Questions:

1. How do you verify that a function and its inverse are correct?
2. What is the domain and range of the inverse function $f^{-1}(x)$?
3. What happens to the graph of a function when finding its inverse?
4. How do you handle finding inverses for non-one-to-one functions?
5. How can you write a function's inverse explicitly if it involves logarithms or more complex exponents?

Tip:

When finding an inverse, always ensure the function is one-to-one (no repeated outputs for different inputs). You can test this using the horizontal line test!