Question

The period of the function:
\[f(x) = \left\{ {\begin{array}{*{20}{c}}
{1,}&{x{\text{ is rational}}}\\
{0,}&{x{\text{ is irrational}}}
\end{array}\;\;\;\;\;is} \right.\]

A.1
B.2
C.non-periodic
D.periodic but having no fundamental period

Answer 1

Hint: A periodic function is a function that repeats its values at regular intervals and constant function is a periodic function with any fundamental period. The horizontal distance required for the graph of a periodic function to complete one cycle. Formally, a function f is periodic if there exists a number p such that f(x + p) = f(x) for all x. The smallest possible value of p is the period. The reciprocal of period is frequency.

Complete step-by-step answer:
Given that:
\[f(x) = \left\{ {\begin{array}{*{20}{c}}
{1,}&{x{\text{ is rational}}}\\
{0,}&{x{\text{ is irrational}}}
\end{array}\;\;\;\;\;is} \right.\]
We know there will be infinite points between two numbers, whether the numbers are irrational or rational.
Hence there would be an irrational number in the immediate neighborhoods of a rational number and there would be rational numbers in the immediate neighborhood of an irrational number.
Hence the function will take the value of 1 and 0 periodically.
However, since there are infinite points on the numbers lines and still infinite points between any two numbers, the period of the function cannot be determined.
Hence, we also know that a constant function is a period function without any fundamental period.
Therefore the answer is the D option.

Note: In this type of question, when periodic function is asked, we need to know how periodic function is determined on any equation. High tides and low tides can be modeled and predicted using periodic functions because scientists can determine the height of the water at different times of the day (when the water level is low, the tide is low).