{"id":3381,"date":"2021-07-23T08:47:51","date_gmt":"2021-07-23T08:47:51","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3381"},"modified":"2021-11-24T16:42:40","modified_gmt":"2021-11-24T11:12:40","slug":"odd-even-function","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/odd-even-function\/","title":{"rendered":"How to Determine Odd Even Function"},"content":{"rendered":"

Here, you will learn what is odd even functions and how to determine if given function is odd or even with example.<\/p>\n

Let’s begin –<\/p>\n

Odd Even Function :<\/h2>\n

Let a function f(x) such that both x and -x are in its domain then<\/p>\n

\n

If f(-x) = f(x) then f is said to be an even function.<\/p>\n

If f(-x) = -f(x) then f is said to be an odd function.<\/p>\n<\/blockquote>\n

Properties of Odd Even Functions :<\/h2>\n

(i) f(x) – f(-x) = 0 \\(\\implies\\) f(x) is even & f(x) + f(-x) = 0 \\(\\implies\\) f(x) is odd.<\/p>\n

(ii) A function may be neither be odd nor even.<\/p>\n

(iii) The only function which is defined on the entire number line & is even and odd at the same time is f(x) = 0.<\/p>\n

(iv) Every constant function is even function.<\/p>\n

(v) Inverse of an even function is not defined.<\/p>\n

(vi) Every even function is symmetric about the y-axis & every odd function is symmetric about origin.<\/p>\n

(vii) If a function is defined as f(a + x) = f(a – x) then this function is symmetric about the line x = a.<\/p>\n\n\n

Example : <\/span>Identify the given functions as odd, even or neither :

\n(i) f(x) = \\(x\\over {e^x – 1}\\) + \\(x\\over 2\\) + 1

\n(ii) f(x + y) = f(x) + f(y) for all x, y \\(\\in\\) R<\/p>\n

Solution : <\/span>(i) f(x) = \\(x\\over {e^x – 1}\\) + \\(x\\over 2\\) + 1

\nClearly domain of f(x) is R – {0}, We have

\nf(-x) = \\(-x\\over {e^{-x} – 1}\\) – \\(x\\over 2\\) + 1

\n= \\(-e^{x}.x\\over {1 – e^x}\\) – \\(x\\over 2\\) + 1

\n= x + \\(x\\over {e^x – 1}\\) – \\(x\\over 2\\) + 1 = \\(x\\over {e^x – 1}\\) + \\(x\\over 2\\) + 1 = f(x)

\nHence, f(x) is an even.

\n(ii) f(x + y) = f(x) + f(y) for all x, y \\(\\in\\) R

\nReplacing x, y by zero, we get f(0) = 2f(0) \\(\\implies\\) f(0) = 0

\nReplacing y by -x, we get f(x) + f(-x) = f(0) = 0 \\(\\implies\\) f(x) = -f(-x)

\nHence, f(x) is an odd.

\n<\/p>\n\n\n

Hope you learnt how to determine if given function is odd or even, learn more concepts of function and practice more questions to get ahead in the competition. Good luck!<\/p>\n\n\n

\n
Next – What is a Periodic Function<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Domain and Range of Modulus Function<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn what is odd even functions and how to determine if given function is odd or even with example. Let’s begin – Odd Even Function : Let a function f(x) such that both x and -x are in its domain then If f(-x) = f(x) then f is said to be an …<\/p>\n

How to Determine Odd Even Function<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[26],"tags":[359,382],"yoast_head":"\nHow to Determine Odd Even Function - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn what is odd function and even function and how to determine odd even function with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/odd-even-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How to Determine Odd Even Function - 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