1.56 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 $1 \leqslant x<\infty$.
1.56 Since

\[
\begin{aligned}
& \sqrt{x+2 \sqrt{x-1}}=\sqrt{x-1+2 \sqrt{x-1}+1}= \\
& =\sqrt{(\sqrt{x-1}+1)^{2}}=\sqrt{x-1}+1 \\
& \sqrt{x-2 \sqrt{x-1}}=|\sqrt{x-1}-1|
\end{aligned}
\]

the given expression will take the form

\( y=\sqrt{x-1}+1+|\sqrt{x-1}-1| \).

Thus, we obtain:

1) when \(\sqrt{x-1}-1<0 \Rightarrow x \in[1,2) \Rightarrow y=2\);
2) when \(\sqrt{x-1}-1 \geqslant 0 \Rightarrow x \in[2, \infty) \Rightarrow y=2 \sqrt{x-1}\).

The graph of the function \( y \) is shown in Fig. 1.1.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-039.jpg?height=481&width=625&top_left_y=969&top_left_x=272)

Fig. 1.1