find the inverse $$ f^{-1}\text{ } $$of $$ f\left(x\right)=\frac{1}{x}-5 $$

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

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### 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:
\[
f^{-1}(x) = \frac{1}{x + 5}
\]

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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.

find the inverse $$ f^{-1} $$of $$ f(x)=\frac{1}{x}-5 $$

Learn how to find the inverse of the function f(x) = 1/x - 5 with a step-by-step solution. This guide covers the steps for solving for the inverse function, including swapping variables, isolating y, and simplifying. Suitable for students in Grades 9-11 studying inverse functions.

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