https://file.aiquickdraw.com/mathbot/file/1726626893981dc60zwzy.jpg

The function provided in the image is \( f(x) = \frac{1}{x+8} \). To find the inverse function, \( f^{-1}(x) \), we need to swap \( x \) and \( y \) in the function, and solve for \( y \):

1. Let \( y = f(x) = \frac{1}{x+8} \).
2. Swap \( x \) and \( y \): \( x = \frac{1}{y+8} \).
3. Solve for \( y \):

\[
y + 8 = \frac{1}{x} \quad \Rightarrow \quad y = \frac{1}{x} - 8
\]

Thus, the inverse function is:

\[
f^{-1}(x) = \frac{1}{x} - 8
\]

You can check the correctness of this inverse function by verifying that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Would you like to go through the verification or have any other questions on this topic? Here are a few more related questions:

1. What would be the domain and range of the inverse function?
2. How does the graph of \( f(x) \) compare to its inverse?
3. What transformations are involved when finding the inverse of a function?
4. How would this process change if the function included a quadratic term instead of a linear term?
5. Why is the subtraction of 8 important in determining the inverse function?

**Tip:** Always verify the inverse by substituting back into the original function and checking if the output is \( x \).

Find the inverse function of f(x) = 1 / (x + 8).

Learn how to find the inverse of the function f(x) = 1 / (x + 8) with a step-by-step solution. This algebra problem covers the process of swapping variables and solving for the inverse function. Suitable for students in Grades 10-12.

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