Problem: The set of all real numbers xxx for which
x+x2+1−1x+x2+1x+\sqrt{x^{2}+1}-\dfrac{1}{x+\sqrt{x^{2}+1}} x+x2+1−x+x2+11
is a rational number is the set of all
Answer Choices:
A. integers xxx
B. rational xxx
C. real xxx
D. xxx for which x2+1\sqrt{x^{2}+1}x2+1 is rational
E. xxx for which x+x2+1x+\sqrt{x^{2}+1}x+x2+1 is rational
Solution:
Note that 1x+x2+1⋅x−x2+1x−x2+1=x2+1−x1\dfrac{1}{x+\sqrt{x^{2}+1}} \cdot \dfrac{x-\sqrt{x^{2}+1}}{x-\sqrt{x^{2}+1}}=\dfrac{\sqrt{x^{2}+1}-x}{1}x+x2+11⋅x−x2+1x−x2+1=1x2+1−x. Hence
x+x2+1−1x+x2+1=2x,x+\sqrt{x^{2}+1}-\dfrac{1}{x+\sqrt{x^{2}+1}}=2 x, x+x2+1−x+x2+11=2x,
which is rational if and only if xxx is rational. To show that this is the only answer, show that each of the other sets is different from the set of rational numbers.