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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
- Sine function (sin(x)): The sine function is a periodic function that oscillates between -1 and 1 over the interval of 2?. It repeats its values every 2? units, so the period of the sine function is 2?. The graph of the sine function is a smooth, wave-like curve.
- Cosine function (cos(x)): The cosine function is also a periodic function that oscillates between -1 and 1 over the interval of 2?. It has the same period as the sine function, which is 2?. The graph of the cosine function is similar to that of the sine function but shifted horizontally.
- Tangent function (tan(x)): The tangent function is periodic with a period of ?. It exhibits vertical asymptotes at regular intervals, causing it to repeat its values every ? units. The tangent function is defined as the ratio of the sine function to the cosine function.
- $f(x) = x – [x]$ 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 2?/|B|, where ‘B’ is the coefficient of ‘x’.
(b) f(x) = sin(Bx + C) or f(x) = cos(Bx + C), the period remains 2?/|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, fg ,f/g is periodic if there exists an LCM of p and q
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 $\pi$
