{"id":13884,"date":"2020-12-10T07:14:30","date_gmt":"2020-12-10T01:44:30","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=13884"},"modified":"2020-12-10T14:53:47","modified_gmt":"2020-12-10T09:23:47","slug":"function","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/function\/","title":{"rendered":"What is a Function?"},"content":{"rendered":"<h2>What is a Function?<\/h2>\n<p><iframe loading=\"lazy\" title=\"What is a Function? (GMAT\/GRE\/CAT\/Bank PO\/SSC CGL) | Don&#039;t Memorise\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/9w8VnZWr8tg?start=1&#038;feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n<h3>Definition of function<\/h3>\n<p><strong>Function<\/strong> can be easily defined with the help of the concept of mapping. Let X and Y be any two non-empty sets. \u201cA function from X to Y is a rule or correspondence that assigns to each element of set X, one and only one element of set Y\u201d. Let the correspondence be \u2018f\u2019 then mathematically we write f : X \u27f6 Y where y = f(x), x \u2208 X and y \u2208 Y. We say that \u2018y\u2019 is the image of \u2018x\u2019 under f (or x is the pre image of y).<br \/>\nTwo things should always be kept in mind:<\/p>\n<ol>\n<li>A mapping f : X \u27f6 Y\u00a0is said to be a function if each element in the set X has its image in set Y. It is also possible that there are few elements in set Y which are not the images of any element in set X.<\/li>\n<li>Every element in set X should have one and only one image. That means it is impossible to have more than one image for a specific element in set X. Functions can not be multi-valued (A mapping that is multi-valued is called a relation from X and Y) e.g.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129466 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/What-is-a-Function-1.jpg\" alt=\"What is a Function 1\" width=\"569\" height=\"308\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/What-is-a-Function-1.jpg 569w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/What-is-a-Function-1-300x162.jpg 300w\" sizes=\"(max-width: 569px) 100vw, 569px\" \/><\/li>\n<\/ol>\n<h3>Testing for a function by vertical line test<\/h3>\n<p>A relation f : A\u00a0\u27f6 B\u00a0is a function or not it can be checked by a graph of the relation. If it is possible to draw a vertical line which cuts the given curve at more than one point then the given relation is not a function and when this vertical line means line parallel to Y-axis cuts the curve at only one point then it is a function. Figure (iii) and (iv) represents a function.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129467 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/What-is-a-Function-2.jpg\" alt=\"What is a Function 2\" width=\"551\" height=\"379\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/What-is-a-Function-2.jpg 551w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/What-is-a-Function-2-300x206.jpg 300w\" sizes=\"(max-width: 551px) 100vw, 551px\" \/><\/p>\n<h3>Number of functions<\/h3>\n<p>Let X and Y be two finite sets having m and n elements respectively. Then each element of set X can be associated to any one of n elements of set Y. So, total number of functions from set X to set Y is n<sup>m<\/sup>.<\/p>\n<h3>Value of the function<\/h3>\n<p>If y = f(x) is a function then to find its values at some value of x, say x = a we directly substitute x = a in its given rule f(x) and it is denoted by f(a).<br \/>\ne.g. If f(x) = x<sup>2<\/sup> + 1, then f(1) = 1<sup>2<\/sup> + 1 = 2, f(2) = 2<sup>2<\/sup> + 1 = 5, f(0) = 0<sup>2<\/sup> + 1 = 1, etc.<\/p>\n<h3>Algebra of functions<\/h3>\n<ol>\n<li><strong>Scalar multiplication of a function:<\/strong><br \/>\n(c f)x = c f(x) where c is a scalar. The new function has the domain\u00a0X<sub>f<\/sub>.<\/li>\n<li><strong>Addition\/subtraction of functions:<\/strong><br \/>\n(f \u00b1 g)(x) = f(x) \u00b1 g(x). The new function has the domain X.<\/li>\n<li><strong>Multiplication of functions:<\/strong><br \/>\n(f.g)(x) = (g.f)(x) = f(x)g(x). The product function has the domain X.<\/li>\n<li><strong>Division of functions:<\/strong><br \/>\n(i) \\(\\left( \\frac { f }{ g } \u00a0\\right) (x)=\\frac { f(x) }{ g(x) }\\). The new function has the domain X, except for the values of x for which g(x) = 0.<br \/>\n(ii) \\(\\left( \\frac { g }{ f } \u00a0\\right) (x)=\\frac { g(x) }{ f(x) }\\). The new function has the domain X, except for the values of x for which f(x) = 0.<\/li>\n<li><strong>Equal functions:<\/strong> Two function f and g are said to be equal functions, if and only if<br \/>\n(i) Domain of f = Domain of g.<br \/>\n(ii) Co-domain of f = Co-domain of g.<br \/>\n(iii) f(x) = g(x) \u2200 x \u2208 their common domain.<\/li>\n<li><strong>Real valued function:<\/strong> If R, be the set of real numbers and A, B are subsets of R, then the function f : A\u00a0\u27f6 B\u00a0is called a real function or real \u2013valued function.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>What is a Function? Definition of function Function can be easily defined with the help of the concept of mapping. Let X and Y be any two non-empty sets. \u201cA function from X to Y is a rule or correspondence that assigns to each element of set X, one and only one element of set &#8230; <a title=\"What is a Function?\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/function\/\" aria-label=\"More on What is a Function?\">Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[5219,5213,5249,5217,5218,5215,5214,5216],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v18.4.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is a Function? - CBSE Library<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cbselibrary.com\/function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is a Function?\" \/>\n<meta property=\"og:description\" content=\"What is a Function? 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