1. We start with the function $f(x) = x + \sqrt{x^{2} - 4}$. We want to find its inverse function $f^{-1}(x)$. 2. To find the inverse, we set $y = x + \sqrt{x^{2} - 4}$ and solve for $x$ in terms of $y$. 3. Isolate the square root term: $$y - x = \sqrt{x^{2} - 4}$$ 4. Square both sides to eliminate the square root: $$ (y - x)^{2} = x^{2} - 4 $$ 5. Expand the left side: $$ y^{2} - 2xy + x^{2} = x^{2} - 4 $$ 6. Subtract $x^{2}$ from both sides: $$ y^{2} - 2xy = -4 $$ 7. Rearrange to isolate terms with $x$: $$ -2xy = -4 - y^{2} $$ 8. Divide both sides by $-2y$ (assuming $y \neq 0$): $$ x = \frac{-4 - y^{2}}{-2y} $$ 9. Simplify the fraction: $$ x = \frac{4 + y^{2}}{2y} $$ 10. Therefore, the inverse function is: $$ f^{-1}(x) = \frac{x^{2} + 4}{2x} $$ 11. Important note: The domain and range must be considered to ensure the inverse is valid. For $f(x)$, the domain is $x \geq 2$ or $x \leq -2$ because of the square root. 12. The inverse function $f^{-1}(x)$ is defined for $x$ in the range of $f$, which is $x \geq 2$ for the principal branch. Final answer: $$ f^{-1}(x) = \frac{x^{2} + 4}{2x} $$