To find the inverse function, $f^{-1}(x)$, of the given function:

$f(x) = \frac{5}{x} + 5$

Step-by-step process:

1. Start by replacing $f(x)$ with $y$:

$y = \frac{5}{x} + 5$

2. Swap $x$ and $y$ to find the inverse:

$x = \frac{5}{y} + 5$

3. Solve for $y$:
   • Subtract 5 from both sides:

$x - 5 = \frac{5}{y}$

• Multiply both sides by $y$:

$y(x - 5) = 5$

• Solve for $y$ by dividing both sides by $x - 5$:

$y = \frac{5}{x - 5}$

Thus, the inverse function is:

$f^{-1}(x) = \frac{5}{x - 5}$

Would you like further details or have any questions?

Here are 5 related questions for practice:

1. If $f(x) = \frac{5}{x+1} + 2$, what is $f^{-1}(x)$?
2. What is the domain of $f^{-1}(x) = \frac{5}{x - 5}$?
3. How can you verify that $f(f^{-1}(x)) = x$ for the function found?
4. If $f(x) = \frac{a}{x} + b$, how do you generally find $f^{-1}(x)$?
5. What are the intercepts of the inverse function $f^{-1}(x)$?

Tip: To find the inverse of a function, swap $x$ and $y$ and then solve for $y$.