1. The problem is to simplify the function $f(x) = x + 2 - 2\sqrt{x - 1}$.\n\n2. We start by recognizing that the function contains a square root term $\sqrt{x - 1}$. The domain of this function requires $x - 1 \geq 0$, so $x \geq 1$.\n\n3. To simplify, consider rewriting the expression as $f(x) = x + 2 - 2\sqrt{x - 1}$.\n\n4. Let $t = \sqrt{x - 1}$, then $t^2 = x - 1$, so $x = t^2 + 1$. Substitute back into $f(x)$:\n$$f(t) = (t^2 + 1) + 2 - 2t = t^2 + 3 - 2t.$$\n\n5. Rearrange the terms to complete the square:\n$$f(t) = t^2 - 2t + 3 = (t - 1)^2 + 2.$$\n\n6. Substitute back $t = \sqrt{x - 1}$:\n$$f(x) = (\sqrt{x - 1} - 1)^2 + 2.$$\n\n7. This is the simplified form of the function, showing it is always greater than or equal to 2 for $x \geq 1$.\n\nFinal answer: $$f(x) = (\sqrt{x - 1} - 1)^2 + 2.$$