Sign function: sign(x)<\/strong><\/p>\n\n\n\nThe sign function is defined as f(x) = x\/|x| for all nonzero values of x, and f(0) = 0. It is an odd function because f(-x) = -f(x) for all x. This can be seen by considering the cases where x is positive, negative, or zero.<\/p>\n\n\n\n
Properties of odd functions<\/h2>\n\n\n\n
(1) For any function f(x), h(x) =f(x) + f(-x) is an odd function.<\/p>\n\n\n\n
h(-x) = f(-x) + f(x) = -f(x) -f(-x)= -[f(x) + f(-x)]=-h(x)<\/p>\n\n\n\n
(2) The sum or difference of two odd functions is odd<\/p>\n\n\n\n
let f(x)= -f(-x) and g(x) = -g(-x)<\/p>\n\n\n\n
Now h(x) = f(x) + g(x)
h(-x) = f(-x) +g(-x) =-f(x) – g(x)= -h(x)<\/p>\n\n\n\n
(3) The multiple of an odd function is again an odd function.<\/p>\n\n\n\n
let f(x)= -f(-x) and h(x) = n f(x)
then
h(-x) = nf(-x) = -nf(x) = -h(x)<\/p>\n\n\n\n
(4) The product or division of two odd functions is even.<\/p>\n\n\n\n
let f(x)=- f(-x) and g(x) = -g(-x)
Now h(x) = f(x) g(x)
h(-x) = f(-x) g(-x) =f(x) g(x)= h(x)<\/p>\n\n\n\n
(5) The composition of two odd functions is odd<\/p>\n\n\n\n
let Both even f(x)= -f(-x) and g(x) =- g(-x)<\/p>\n\n\n\n
$f \\circ g (x) = f(g(x)$
Now $f \\circ g (-x)= f(g(-x)=- f(g(x)$<\/p>\n\n\n\n
How to Prove the odd Function<\/h2>\n\n\n\n
We simply need to prove that f(x) =- f(-x)<\/p>\n\n\n\n
Solved Examples<\/h2>\n\n\n\n
Example 1<\/strong><\/p>\n\n\n\n$f(x) = x^5 -x^3 -2x$<\/p>\n\n\n\n
Solution<\/strong><\/p>\n\n\n\n$f(-x) =(-x)^5 -(-x)^3 -2(-x)=-x^5 +x^3 +2x= -(x^5 -x^3 -2x) =-f(x)$<\/p>\n\n\n\n
Hence an odd function<\/p>\n\n\n\n
Example 2<\/strong><\/p>\n\n\n\n$f(x)= \\frac { \\sqrt {1+x} + \\sqrt {1-x}}{\\sqrt {2+x} – \\sqrt {2-x}}$<\/p>\n\n\n\n
Solution<\/strong><\/p>\n\n\n\n$f(-x) =\\frac { \\sqrt {1-x} + \\sqrt {1+x}}{\\sqrt {2-x} – \\sqrt {2+x}}= -\\frac { \\sqrt {1+x} + \\sqrt {1-x}}{\\sqrt {2+x} – \\sqrt {2-x}}= -f(x) $<\/p>\n\n\n\n
Hence odd function<\/p>\n","protected":false},"excerpt":{"rendered":"
What is odd functions An odd function is a function f(x) that satisfies the property f(-x) = -f(x) for all x in the domain of the function. In other words, an odd function is symmetric with respect to the origin. Geometrically, an odd function has the property that if you rotate the graph of the […]<\/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":"default","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-7958","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\n
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