Periodic Functions - Practice Questions & MCQ

A function f(x) is called a periodic function, if there exists a  +ve real number T such that f(x+T) = f(x) ∀ x? The domain of f(x).

Here, T is called the period of f(x), where T is the least +ve value.

Graphically: if the graph repeats at a fixed interval, the function is said to be periodic and its period is the width of that interval.

Eg

Graph of sin(x) is repeated at an interval of 2π

Some standard results

Properties of the periodic function

i) if f(x) is periodic with period T, then 

1. cf(x) is periodic with period T

2. f(x+c) is periodic with period T

3. f(x) ± c is periodic with period T, where c is any constant.

ii) if $\mathrm{f}(\mathrm{x})$ is periodic with period T , then $\mathrm{kf}(\mathrm{cx}+\mathrm{d})$ has period $\frac{T}{|c|}$ i.e. period is only affected by the coefficient of x ,
iii) if $f_1(x), f_2(x)$ are periodic functions with periods $T_1, T_2$ respectively, then $h(x)=f_1(x)+f_2(x)$ has period
a). LCM of $\left\{T_1, T_2\right\}$, if $\mathrm{h}(\mathrm{x})$ is not an even function.
or
b) 0.5 LCM of $\left\{T_1, T_{2\}}\right.$, if $f_1(x)$ and $f_2(x)$ are complementary pairwise comparable functions.