$1 \cdot 80$ has a set of real numbers $x$, where $x$ makes the value of $x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}}$ a rational number. Then, the set of all such $x$ is
(A) the set of integers.
(B) the set of rational numbers.
(C) the set of real numbers.
(D) the set of numbers for which $\sqrt{x^{2}+1}$ is rational.
(E) the set of numbers for which $x+\sqrt{x^{2}+1}$ is rational.
(40th American High School Mathematics Examination, 1989)
[Solution] From $\frac{1}{x+\sqrt{x^{2}+1}}$
$$
\begin{array}{l}
=\frac{\sqrt{x^{2}+1}-x}{\left(\sqrt{x^{2}+1}+x\right)\left(\sqrt{x^{2}+1}-x\right)} \\
=\sqrt{x^{2}+1}-x,
\end{array}
$$
$\therefore$ the original expression $=2 x$, so when $x$ is a rational number, the original expression is also a rational number. Therefore, the answer is $(B)$.