2.285. Simplify the expression $y=\sqrt{x+2 \sqrt{x-1}}+\sqrt{x-2 \sqrt{x-1}}$, and then plot the graph of the function $y$ for $1 \leq x<\infty$.
Solution.

Domain of definition: $x \geq 0$.

\[
\begin{aligned}
& y=\sqrt{x+2 \sqrt{x-1}}+\sqrt{x-2 \sqrt{x-1}} z \\
& =\sqrt{x-1+2 \sqrt{x-1}+1}+\sqrt{x-1-2 \sqrt{x-1}+1}= \\
& =\sqrt{(\sqrt{x-1}+1)^{2}}+\sqrt{(\sqrt{x-1}-1)^{2}}=\sqrt{x-1}+1+|\sqrt{x-1}-1|= \\
& =\left\{\begin{array}{l}
\sqrt{x-1}+1-\sqrt{x-1}+1=2, \text { if } 1 \leq x \leq 2 \\
\sqrt{x-1}+1+\sqrt{x-1}-1=2 \sqrt{x-1}, \text { if } x>2
\end{array}\right.
\end{aligned}
\]

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0118.jpg?height=787&width=1149&top_left_y=1145&top_left_x=-1)

Answer: $y=\left\{\begin{array}{l}2 \text { if } 1 \leq x \leq 2 \\ 2 \sqrt{x-1} \text { if } x>2 .\end{array}\right.$