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Periodic function

A periodic function is a mathematical function that repeats its values in a predictable pattern over a specific interval. This interval is known as the period of the function. In other words, if you were to graph a periodic function, you would observe the same pattern repeating at regular intervals

Definition

A function is said to be periodic function if there exists a positive real number p such that

f(x+p) = f(x) , $x \in D$

The least of all such positive number p is called the fundamental period of function. And this is the called the period of the function

This means that if you shift the input variable x by the period p, the function’s values remain the same.

Examples

1. Sine function (sin(x)): The sine function is a periodic function that oscillates between -1 and 1 over the interval of $2 \pi$. It repeats its values every $2 \pi$ units, so the period of the sine function is $2\pi$. The graph of the sine function is a smooth, wave-like curve.
2. Cosine function (cos(x)): The cosine function is also a periodic function that oscillates between -1 and 1 over the interval of $2 \pi$. It has the same period as the sine function, which is $2 \pi$. The graph of the cosine function is similar to that of the sine function but shifted horizontally.
3. Tangent function (tan(x)): The tangent function is periodic with a period of $\pi$. It exhibits vertical asymptotes at regular intervals, causing it to repeat its values every $\pi$ units. The tangent function is defined as the ratio of the sine function to the cosine function.
4. $f(x) = x – [x]$ is also a periodic function with the period being 1

These are just a few examples of periodic functions, but there are many other periodic functions with different periods and properties. It’s worth noting that not all functions are periodic. For a function to be periodic, its values must repeat in a predictable manner over a specific interval.

Rules for finding the period of the periodic functions

(1) If f(x) is a periodic function with period p, the $a f(x) +b$ where a , b are real numbers and a is not zero is also a periodic function of period p

Example

f(x) = 2 sin(x) + 3

We know that sin (x) is a periodic function with period $\2 pi$, so this is also a periodic function with period $2\pi$

(2)If f(x) is a periodic function with period p, the $f(ax +b)$ where a , b are real numbers and a is not zero is also a periodic function of period p/|a|. This is specially useful for complex trigonometric functions

Example

(a)f(x) = sin(Bx), the period is $\frac {2 \pi}{|B|}$, where ‘B’ is the coefficient of ‘x’.

(b) f(x) = sin(Bx + C) or f(x) = cos(Bx + C), the period remains $\frac {2 \pi}{|B|}$, because the ‘C’ only shifts the graph left or right.

(3) If f(x) and g(x) are two periodic functions with period p and q respectively, then the function $f + g$, $f-g$, $f.g$ ,$\frac {f}{g}$ is periodic if there exists an LCM of p and q. And there does not exists any number m < LCM for  f(x+m) + g(x+m) =f(x) + g(x), if its exists then m is the Period

Example

h(x) = sin 2x + cos 4x

Here sin 2x is periodic function with period $\pi$
cos 4x is a periodic function with period $\pi/2$

Now LCM is $\pi$

So, it is a periodic function with period $\pi$

Practice Questions

Question 1

Find the period of the function
(a)f(x) = sin(2x).
(b)$g(x) = cos(3x + \pi/4)$.
(c)h(x) = 2sin(0.5x) + 1.
(d)k(x) = cos(x/3).

Solutions

(a)f(x) = sin(2x): The period is $2 \pi /|B|$. Here, B = 2. Therefore, the period is $2 \pi /2 = \pi$.
(b)$g(x) = cos(3x + \pi/4)$: The period is $2\pi/|B|$. Here, B = 3. Therefore, the period is $2\pi/3$.
(c)h(x) = 2sin(0.5x) + 1: The period is $2\pi/|B|$. Here, B = 0.5. Therefore, the period is $2\pi/0.5 = 4\pi$.
(d)k(x) = cos(x/3): The period is $2\pi/|B|$. Here, B = 1/3. Therefore, the period is $2\pi/(1/3) = 6\pi$

Question 2

Find the period of the function |sin x|

Solutions

The sin(x) function has a period of $2\pi$. However, when we take the absolute value, the negative half of the cycle (from $\pi$ to $2\pi$) gets reflected to become positive, essentially duplicating the first half of the cycle (from 0 to $\pi$).

Therefore, the period of the |sin(x)| function is $\pi$

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