1. **State the problem:** Find the inverse function of $f(x) = \frac{1}{x+5}$ where $x \neq -5$. 2. **Recall the formula for inverse functions:** To find the inverse, swap $x$ and $y$ in the equation $y = f(x)$ and solve for $y$. 3. **Start with the original function:** $$y = \frac{1}{x+5}$$ 4. **Swap $x$ and $y$:** $$x = \frac{1}{y+5}$$ 5. **Solve for $y$:** Multiply both sides by $y+5$: $$x(y+5) = 1$$ 6. **Distribute $x$:** $$xy + 5x = 1$$ 7. **Isolate $y$:** $$xy = 1 - 5x$$ 8. **Divide both sides by $x$ (noting $x \neq 0$):** $$y = \frac{1 - 5x}{x}$$ 9. **Simplify the fraction:** $$y = \frac{1}{x} - 5$$ 10. **State the inverse function:** $$f^{-1}(x) = \frac{1}{x} - 5, \quad x \neq 0$$ 11. **Check domain restrictions:** Since $x \neq -5$ in the original function, the inverse function has $x \neq 0$ to avoid division by zero. **Final answer:** The inverse function is $y = \frac{1}{x} - 5$ with $x \neq 0$, which corresponds to option A.