To find the inverse of the function $f(x) = \frac{1}{x + 3}$, follow these steps:

Step 1: Replace $f(x)$ with $y$

$y = \frac{1}{x + 3}$

Step 2: Swap $x$ and $y$

$x = \frac{1}{y + 3}$

Step 3: Solve for $y$

Multiply both sides by $y + 3$: $x(y + 3) = 1$

Distribute $x$: $xy + 3x = 1$

Isolate $y$: $xy = 1 - 3x$

Finally, solve for $y$ by dividing by $x$: $y = \frac{1 - 3x}{x}$

Step 4: Rewrite the inverse function

So, the inverse of $f(x) = \frac{1}{x + 3}$ is: $f^{-1}(x) = \frac{1 - 3x}{x}$

Would you like any further details on this, or have any questions? Here are 5 questions that expand on the concept:

1. What are the domain and range of $f(x)$ and its inverse $f^{-1}(x)$?
2. How do we verify if two functions are inverses of each other?
3. What happens to the graph of $f(x)$ when we graph its inverse?
4. Can inverse functions be found for all types of functions?
5. How do restrictions on the domain affect the inverse function?

Tip: Always ensure the original function is one-to-one before attempting to find its inverse.