Question
Which of the following functions is periodic?
A.
$$f\left( x \right) = x - \left[ x \right]$$ where $$\left[ x \right]$$ denotes the largest integer less than or equal to the real number $$x$$
B.
$$f\left( x \right) = \sin \frac{1}{x}\,{\text{for}}\,x \ne 0,f\left( 0 \right) = 0$$
C.
$$f\left( x \right) = x\cos x$$
D.
none of these
Answer :
$$f\left( x \right) = x - \left[ x \right]$$ where $$\left[ x \right]$$ denotes the largest integer less than or equal to the real number $$x$$
Solution :
\[f\left( x \right) = x - \left[ x \right] = \left\{ {\begin{array}{*{20}{c}} { \ldots .}\\ {\begin{array}{*{20}{c}} {x - 1,}\\ {\begin{array}{*{20}{c}} {x - 2,}\\ {x - 3,} \end{array}} \end{array}}\\ { \ldots .} \end{array}\,\,\begin{array}{*{20}{c}} {1 \le x < 2}\\ {2 \le x < 3}\\ {3 \le x < 4} \end{array}} \right.\]
graph of function is
Clearly it is a periodic function with period 1.
$$\therefore \left( a \right)$$ is the correct alternative.
\[f\left( x \right) = x - \left[ x \right] = \left\{ {\begin{array}{*{20}{c}} { \ldots .}\\ {\begin{array}{*{20}{c}} {x - 1,}\\ {\begin{array}{*{20}{c}} {x - 2,}\\ {x - 3,} \end{array}} \end{array}}\\ { \ldots .} \end{array}\,\,\begin{array}{*{20}{c}} {1 \le x < 2}\\ {2 \le x < 3}\\ {3 \le x < 4} \end{array}} \right.\]
graph of function is
Clearly it is a periodic function with period 1.
$$\therefore \left( a \right)$$ is the correct alternative.