$\begin{array}{l} A = \sqrt {x + \sqrt {2x - 1} } - \sqrt {x - \sqrt {2x - 1} } \\ \Rightarrow A\sqrt 2 = \sqrt {2x + 2\sqrt {2x - 1} } - \sqrt {2x - 2\sqrt {2x - 1} } \\ \Rightarrow A\sqrt 2 = \sqrt {{{\left( {\sqrt {2x - 1} + 1} \right)}^2}} - \sqrt {{{\left( {\sqrt {2x - 1} - 1} \right)}^2}} \\ \Rightarrow A\sqrt 2 = \left| {\sqrt {2x - 1} + 1} \right| - \left| {\sqrt {2x - 1} - 1} \right|\\ \Leftrightarrow \left[ \begin{array}{l} A\sqrt 2 = 2 \Rightarrow A = \sqrt 2 \left( {x \ge 1} \right)\\ A\sqrt 2 = 2\sqrt {2x - 1} \Rightarrow A = \sqrt 2 \sqrt {2x - 1} \left( {x \le 1} \right) \end{array} \right. \end{array}$
