To find the inverse function $f^{-1}(x)$ for $f(x) = \frac{1}{x} - 5$, follow these steps:

---

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

Let $y = f(x)$. This gives: $y = \frac{1}{x} - 5$

---

Step 2: Swap $x$ and $y$

For the inverse, interchange $x$ and $y$: $x = \frac{1}{y} - 5$

---

Step 3: Solve for $y$

Rearrange the equation to isolate $y$:

1. Add 5 to both sides: $x + 5 = \frac{1}{y}$

2. Take the reciprocal of both sides: $y = \frac{1}{x + 5}$

---

Step 4: Write the inverse function

The inverse function is: $f^{-1}(x) = \frac{1}{x + 5}$

---

Would you like to see a graph of $f(x)$ and its inverse to verify? Let me know!

Related Questions

1. How do you verify that $f^{-1}(x)$ is correct?
2. Can you find the domain and range of $f(x)$ and $f^{-1}(x)$?
3. What happens to $f^{-1}(x)$ as $x \to -5$?
4. How does the horizontal asymptote of $f(x)$ relate to the vertical asymptote of $f^{-1}(x)$?
5. What is the composition $f(f^{-1}(x))$ and how does it simplify?

Tip

Always check the domain restrictions of the original function when finding its inverse to ensure they match.