To find the inverse of the function $f(x) = \frac{5 + x}{x + 1}$, we need to follow these steps:

1. Rewrite $f(x)$ as $y$: $y = \frac{5 + x}{x + 1}$

2. Swap $x$ and $y$: This will help us solve for $y$ in terms of $x$ to find the inverse. $x = \frac{5 + y}{y + 1}$

3. Solve for $y$:

   • Multiply both sides by $y + 1$ to eliminate the denominator: $x(y + 1) = 5 + y$

   • Expand and rearrange terms: $xy + x = 5 + y$

   • Get all terms involving $y$ on one side of the equation: $xy - y = 5 - x$

   • Factor out $y$ on the left side: $y(x - 1) = 5 - x$

   • Solve for $y$: $y = \frac{5 - x}{x - 1}$

4. Write the inverse function: The inverse function $f^{-1}(x)$ is: $f^{-1}(x) = \frac{5 - x}{x - 1}$

Would you like further details on any of these steps, or do you have any questions?

Here are some related questions that build on this solution:

1. What are the domain and range of $f(x) = \frac{5 + x}{x + 1}$ and its inverse?
2. How do we determine if a function has an inverse?
3. How would we verify that $f(f^{-1}(x)) = x$ for this function?
4. Can you explain the significance of horizontal and vertical asymptotes for $f(x)$ and $f^{-1}(x)$?
5. How does the graph of $f(x)$ relate to the graph of its inverse?

Tip: When finding inverses, always ensure the original function is one-to-one, as only one-to-one functions have inverses.