To solve the problem, we will first find the inverse of the function $f(x)$, then compute $f(f^{-1}(x))$.

Step 1: Find the equation for the inverse $f^{-1}(x)$

To find the inverse, follow these steps:

1. Start with $y = f(x)$.
2. Swap $x$ and $y$: $x = f(y)$.
3. Solve for $y$, which gives $f^{-1}(x)$.

The function you are working with is not provided, so I'll explain the general approach. If you have a specific function $f(x)$ in mind, please provide it, and I'll apply the steps to that particular function.

Step 2: Compute $f(f^{-1}(x))$

For any function $f$ and its inverse $f^{-1}$, the identity $f(f^{-1}(x)) = x$ holds true. This means that when you compose a function with its inverse, the result is always the original input $x$.

Thus, $f(f^{-1}(x)) = x$.

Summary:

1. Inverse of $f(x)$: To find the inverse, swap $x$ and $y$ in $f(x) = y$, then solve for $y$.
2. Composition of a function with its inverse: $f(f^{-1}(x)) = x$.

Let me know if you have a specific function for which you need to find the inverse, or if you'd like a more detailed breakdown.

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Would you like a step-by-step solution for a particular function? Here are some examples of related questions:

1. How do I find the inverse of $f(x) = 3x + 2$?
2. How do I solve for $f^{-1}(x)$ when $f(x) = \frac{x-1}{x+2}$?
3. Can you explain the graphical interpretation of $f(f^{-1}(x)) = x$?
4. What is the inverse of an exponential function like $f(x) = e^x$?
5. How does the inverse function work for quadratic functions?

Tip: The inverse of a function essentially "reverses" the effect of the original function. To ensure an inverse exists, the original function must be one-to-one (bijective).