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<\/p>\n\n\n\n
A function is said to be periodic function if there exists a positive real number p such that<\/p>\n\n\n\n
f(x+p) = f(x) , $x \\in D$<\/p>\n\n\n\n
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<\/p>\n\n\n\n
This means that if you shift the input variable x by the period p, the function’s values remain the same.<\/p>\n\n\n\n
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.<\/p>\n\n\n\n
(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<\/p>\n\n\n\n
Example<\/strong><\/p>\n\n\n\n f(x) = 2 sin(x) + 3<\/p>\n\n\n\n We know that sin (x) is a periodic function with period $\\2 pi$, so this is also a periodic function with period $2\\pi$<\/p>\n\n\n\n (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<\/p>\n\n\n\n Example<\/strong><\/p>\n\n\n\n (a)f(x) = sin(Bx), the period is $\\frac {2 \\pi}{|B|}$, where ‘B’ is the coefficient of ‘x’.<\/p>\n\n\n\n (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.<\/p>\n\n\n\n (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<\/p>\n\n\n\n Example<\/strong><\/p>\n\n\n\n h(x) = sin 2x + cos 4x<\/p>\n\n\n\n Here sin 2x is periodic function with period $\\pi$ Now LCM is $\\pi$<\/p>\n\n\n\n So, it is a periodic function with period $\\pi$<\/p>\n\n\n\n Question 1<\/strong><\/p>\n\n\n\n Find the period of the function Solutions<\/strong><\/p>\n\n\n\n (a)f(x) = sin(2x): The period is $2 \\pi \/|B|$. Here, B = 2. Therefore, the period is $2 \\pi \/2 = \\pi$. Question 2<\/strong><\/p>\n\n\n\n Find the period of the function |sin x|<\/p>\n\n\n\n Solutions<\/strong><\/p>\n\n\n\n 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 $\\pi$).<\/p>\n\n\n\n Therefore, the period of the |sin(x)| function is $\\pi$<\/p>\n\n\n\n Hope you like this content on Periodic function in Maths<\/p>\n\n\n\n Related Articles<\/strong><\/p>\n\n\n\n Odd Functions<\/a> 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 of Periodic Function A […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-8111","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\n
cos 4x is a periodic function with period $\\pi\/2$<\/p>\n\n\n\nPractice Questions on Periodic Function<\/h2>\n\n\n\n
(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).
<\/p>\n\n\n\n
(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 $2\\pi\/(1\/3) = 6\\pi$<\/p>\n\n\n\n
Even functions<\/a>
Into Functions<\/a>
Many one Functions<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"