1989 AJHSME Problems/Problem 18

Problem

Many calculators have a reciprocal key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is depressed, then the display becomes $\boxed{000.25}$ . If $\boxed{00032}$ is currently displayed, what is the fewest number of times you must depress the $\boxed{\frac{1}{x}}$ key so the display again reads $\boxed{00032}$ ?

Let $f(x)=\frac{1}{x}$ . We have $f(f(x))=\frac{1}{\frac{1}{x}}=x$ Thus, we need to iterate the key pressing twice to get the display back to the original $\rightarrow \boxed{\text{B}}$ .