To find the inverse $f^{-1}$ of a function $f(x)$ and determine its range, we need to have a complete definition of $f(x)$, particularly the expression on the right side of $f(x) = \ldots$.

Without the expression, it's difficult to provide an exact answer, but I can guide you through the general steps you would take:

1. Identify the Expression: Start by ensuring we know the exact formula of $f(x)$ given on the right side of the equation.

2. Set $y = f(x)$: To find the inverse function, set $y = f(x)$ and solve for $x$ in terms of $y$.

3. Solve for $x$ in terms of $y$: Rearrange the equation to express $x$ as a function of $y$. This new expression will represent $f^{-1}(y)$.

4. Determine the Range of $f$: Since the domain of $f$ is $\mathbb{R} - \{2\}$, you’d need to consider any limitations or asymptotes in $f(x)$ to find its range.

5. Adjust Domain of $f^{-1}$: Ensure the domain of $f^{-1}$ corresponds to the range of $f(x)$.

If you provide the specific formula for $f(x)$, I can give you the exact solution for $f^{-1}$ and determine the range of $f$. Would you like further clarification on these steps?

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Here are five related questions for exploration:

1. How can we find the domain and range of a function analytically?
2. What conditions must a function satisfy to be invertible?
3. How does removing values from the domain affect the inverse of a function?
4. What are common techniques for solving for the inverse of rational functions?
5. How does one determine if a function is one-to-one?

Tip: When finding an inverse, always verify that the original function is one-to-one over its domain to ensure the inverse exists.