{"id":100694,"date":"2022-02-12T17:09:02","date_gmt":"2022-02-12T11:39:02","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=100694"},"modified":"2024-02-23T16:22:03","modified_gmt":"2024-02-23T10:52:03","slug":"relations-and-functions","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/","title":{"rendered":"Relations and Functions"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Domain_Co%E2%88%92Domain_Range_Of_A_Function\" title=\"Domain, Co\u2212Domain &amp; Range Of A Function\">Domain, Co\u2212Domain &amp; Range Of A Function<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Important_Types_Of_Functions\" title=\"Important Types Of Functions \n\">Important Types Of Functions \n<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Domains_And_Ranges_Of_Common_Function\" title=\"Domains And Ranges Of Common Function\">Domains And Ranges Of Common Function<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Equal_Or_Identical_Function_Relations_and_Functions\" title=\"Equal Or Identical Function | Relations and Functions\">Equal Or Identical Function | Relations and Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Classification_Of_Functions\" title=\"Classification Of Functions\">Classification Of Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Algebraic_Operations_On_Functions\" title=\"Algebraic Operations On Functions\">Algebraic Operations On Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Composite_Of_Uniformly_Non-Uniformly_Defined_Functions\" title=\"Composite Of Uniformly &amp; Non-Uniformly Defined Functions\">Composite Of Uniformly &amp; Non-Uniformly Defined Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Homogeneous_Functions_Relations_and_Functions\" title=\"Homogeneous Functions | Relations and Functions\">Homogeneous Functions | Relations and Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Bounded_Function_Relations_and_Functions\" title=\"Bounded Function | Relations and Functions\">Bounded Function | Relations and Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Implicit_Explicit_Function_Relations_and_Functions\" title=\"Implicit &amp; Explicit Function | Relations and Functions\">Implicit &amp; Explicit Function | Relations and Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Inverse_of_A_Function_Relations_and_Functions\" title=\"Inverse of A Function | Relations and Functions\">Inverse of A Function | Relations and Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Odd_and_Even_Functions_Relations_and_Functions\" title=\"Odd and Even Functions | Relations and Functions\">Odd and Even Functions | Relations and Functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/cbse-notes\/class-11\/maths\/relations-and-functions\/#Periodic_Function_Relations_and_Functions\" title=\"Periodic Function | Relations and Functions\">Periodic Function | Relations and Functions<\/a><\/li><\/ul><\/nav><\/div>\n<p><span style=\"font-size: 14pt;\"><strong>Relations and Functions<\/strong><\/span><\/p>\n<p>If to every value of x belonging to some set E there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on E.Conventionally the word \u201cFUNCTION\u201d is used only as the meaning of a single valued function, if not otherwise stated. Pictorially: \\(\\xrightarrow [ x ]{ input } \\boxed { f }\\)<br \/>\n\\(\\frac{\\mathrm{f}(\\mathrm{x})=\\mathrm{y}}{\\text { output }}\\) y is called the image of x &amp; x is the pre-image of y under f. Every function from A \u2192 B<br \/>\nsatisfies the following conditions.<\/p>\n<ul>\n<li>f \u2282 A x B<\/li>\n<li>\u2200 a \u2208 A \u21d2 (a, f(a)) \u2208 f and<\/li>\n<li>(a, b) \u2208 f &amp; (a, c) \u2208 f \u21d2 b = c<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Domain_Co%E2%88%92Domain_Range_Of_A_Function\"><\/span>Domain, Co\u2212Domain &amp; Range Of A Function<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Let f : A \u2192 B, then the set A is known as the domain of f &amp; the set B is known as co-domain of f. The set of all f images of elements of A is known as the range of f. Thus<\/p>\n<p>Domain of f = { a | a \u2208 A, (a, f(a)) \u2208 f} Range of f = { f(a) | a \u2208 A, f(a) \u2208 B}<br \/>\nIt should be noted that range is a subset of co\u2212domain. If only the rule of function is given then the domain of the function is the set of those real numbers, where function is defined. For a continuous function, the interval from minimum to maximum value of a function gives the range.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Important_Types_Of_Functions\"><\/span>Important Types Of Functions<strong><br \/>\n<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li><strong>Polynomial Function:<\/strong><br \/>\nIf a function f is defined by f (x) = a<sub>0<\/sub>x<sup>n<\/sup> + a<sub>1<\/sub>x<sup>n-1<\/sup> + a<sub>2<\/sub>x<sup>n-2<\/sup> + \u2026 + a<sub>n-1<\/sub>x + a<sub>n<\/sub> where n is a non negative integer and a<sub>0<\/sub>, a<sub>1<\/sub>, a<sub>2<\/sub>, \u2026, a<sub>n<\/sub> are real numbers and a<sub>0<\/sub> \u2260 0, then f is called a polynomial function of degree n<br \/>\n<strong>Note:<br \/>\n<\/strong>(a) A polynomial of degree one with no constant term is called an odd linear function. i.e. f(x) = ax , a \u2260 0<br \/>\n(b) There are two polynomial functions, satisfying the relation; f(x).f(1\/x) = f(x) + f(1\/x). They are:<br \/>\n(i) f(x) = x<sup>n<\/sup> + 1 &amp;<br \/>\n(ii) f(x) = 1 \u2212 x<sup>n<\/sup> , where n is a positive integer.<\/li>\n<li><strong>Algebraic Function:<\/strong> y is an algebraic function of x if it is a function that satisfies an algebraicequation of the form<br \/>\nP<sub>0<\/sub>(x) y<sup>n<\/sup> + P<sub>1<\/sub> (x) y<sup>n-1<\/sup> + \u2026\u2026. + P<sub>n-1<\/sub> (x) y + P<sub>n<\/sub> (x) = 0 Where n is a positive integer and<br \/>\nP<sub>0<\/sub>(x), P<sub>1<\/sub>(x) \u2026\u2026\u2026.. are Polynomials in x.<br \/>\ne.g. y = |x| is an algebraic function, since it satisfies the equation y\u00b2 \u2212 x\u00b2 = 0.<br \/>\nNote that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called Transcedental Function.<\/li>\n<li><strong>Fractional Rational Function:<\/strong> A rational function is a function of the form.<br \/>\n\\(y=f(x)\\frac {g(x)}{h(x)}\\)<br \/>\nwhere g (x) &amp; h (x) are polynomials &amp; h (x) \u2260 0.<\/li>\n<li><strong>Absolute Value Function:<\/strong> A function y = f (x) = |x| is called the absolute value function or Modulus function. It is defined as:<br \/>\n\\(\\mathrm{y}=|\\mathrm{x}|=\\left[ \\begin{array}{ll}{\\mathrm{x}} &amp; {\\text { if } \\quad \\mathrm{x} \\geq 0} \\\\ {-\\mathrm{x}} &amp; {\\text { if } \\quad \\mathrm{x}&lt;0}\\end{array}\\right.\\)<\/li>\n<li><strong>Exponential Function:<\/strong> A function f(x) = a<sup>x<\/sup> = e<sup>xlna<\/sup>  (a &gt; 0 , a \u2260 1, x \u2208 R) is called anexponential function. The inverse of the exponential function is called the logarithmic function . i.e. g(x) = log<sub>a<\/sub>x.<br \/>\nNote that f(x) &amp; g(x) are inverse of each other &amp; their graphs are as shown.<\/li>\n<li><strong>Signum Function:<\/strong><br \/>\nA function y= f (x) = Sgn (x) is defined as follows:<br \/>\n\\(\\mathrm{y}=\\mathrm{f}(\\mathrm{x})=\\left[ \\begin{array}{ccc}{1} &amp; {\\text { for }} &amp; {\\mathrm{x}&gt;0} \\\\ {0} &amp; {\\text { for }} &amp; {\\mathrm{x}=0} \\\\ {-1} &amp; {\\text { for }} &amp; {\\mathrm{x}&lt;0}\\end{array}\\right.\\)<br \/>\nIt is also written as Sgn x = |x|\/ x ; x \u2260 0 ; f (0) = 0<\/li>\n<li><strong>Greatest Integer Or Step-Up Function:<\/strong><br \/>\nThe function y = f (x) = [x] is called the greatest integer function where [x] denotes the greatest integer less than or equal to x. Note that for:<br \/>\n\u2212 1 \u2264 x &lt; 0 ; [x] = \u2212 1 0 \u2264 x &lt; 1 ; [x] = 0<br \/>\n1 \u2264 x &lt; 2 ; [x] = 1 2 \u2264 x &lt; 3 ; [x] = 2 and so on .<br \/>\n<strong>Properties of<\/strong><strong> greatest<\/strong><strong> integer function:<br \/>\n<\/strong><\/p>\n<ul>\n<li>[x] \u2264 x &lt; [x] + 1 and x \u2212 1 &lt; [x] \u2264 x , 0 \u2264 x \u2212 [x] &lt; 1<\/li>\n<li>[x + m] = [x] + m if m is an integer.<\/li>\n<li>[x] + [y] \u2264 [x + y] \u2264 [x] + [y] + 1<\/li>\n<li>[x] + [\u2212 x] = 0 if x is an integer = \u2212 1 otherwise.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Fractional Part Function:<\/strong><br \/>\nIt is defined as:<br \/>\ng (x) = { x} = x \u2212 [x] .<br \/>\ne.g. the fractional part of the no. 2.1 is<br \/>\n2.1\u2212 2 = 0.1 and the fractional part of \u2212 3.7 is 0.3. The period of this function is 1 and graph of this function is as shown.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Domains_And_Ranges_Of_Common_Function\"><\/span>Domains And Ranges Of Common Function<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"Equal_Or_Identical_Function_Relations_and_Functions\"><\/span>Equal Or Identical Function | Relations and Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Two functions f &amp; g are said to be equal if :<\/p>\n<ul>\n<li>The domain of f = the domain of g.<\/li>\n<li>The range of f = the range of g and<\/li>\n<li>f(x) = g(x) , for every x belonging to their common domain.<br \/>\neg. f(x) = \\(\\frac {1}{x}\\) &amp; g(x) = \\(\\frac{x}{x^{2}}\\) are identical functions.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Classification_Of_Functions\"><\/span>Classification Of Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>One \u2212 One Function (Injective mapping):<\/strong><br \/>\nA function f : A \u2192 B is said to be a one\u2212one function or injective mapping if different elements of A have different f images in B. Thus for x<sub>1<\/sub>, x<sub>2<\/sub> \u2208 A &amp; f(x<sub>1<\/sub>), f(x<sub>2<\/sub>) \u2208 B , f(x<sub>1<\/sub>) = f(x<sub>2<\/sub>) \u21d4 x<sub>1<\/sub> = x<sub>2<\/sub> or x<sub>1<\/sub> \u2260 x<sub>2<\/sub> \u21d4 f(x<sub>1<\/sub>) \u2260 f(x<sub>2<\/sub>) .<br \/>\nDiagrammatically an injective mapping can be shown as<br \/>\nNote:<strong><br \/>\n<\/strong><\/p>\n<ul>\n<li>Any function which is entirely increasing or decreasing in whole domain, then f(x) is one\u2212one.<\/li>\n<li>If any line parallel to the x-axis cuts the graph of the function at most at one point, then the function is one\u2212one.<\/li>\n<\/ul>\n<p><strong>Many\u2013one function:<\/strong><br \/>\nA function f : A \u2192 B is said to be a many one function if two or more elements of A have the same  f image in B. Thus f : A \u2192 B is many one if for; x<sub>1<\/sub>, x<sub>2<\/sub> \u2208 A, f(x<sub>1<\/sub>) = f(x<sub>2<\/sub>) but x<sub>1<\/sub> \u2260 x<sub>2<\/sub><br \/>\nDiagrammatically a many one mapping can be shown as<br \/>\nNote:<\/p>\n<ul>\n<li>Any continuous function which has atleast one local maximum or local minimum, then f(x) is many\u2212one. In other words, if a line parallel to x\u2212axis cuts the graph of the function atleast at two points, then f is many\u2212one.<\/li>\n<li>If a function is one\u2212one, it cannot be many\u2212one and vice versa.<\/li>\n<\/ul>\n<p><strong>Onto function (Surjective mapping):<\/strong> If the function f : A \u2192 B is such that each element in B (co\u2212domain) is the f image of atleast one element in A, then we say that f is a function of A \u2018onto\u2019 B. Thus f : A \u2192 B is surjective iff \u2200 b \u2208 B, \u2203 some a \u2208 A such that f (a) = b.<br \/>\nDiagramatically surjective mapping can be shown as<br \/>\nNote: if range = co\u2212domain, then f(x) is onto.<\/p>\n<p><strong>Into function:<\/strong><br \/>\nIf f : A \u2192 B is such that there exists atleast one element in co\u2212domain which is not the image of any element in domain, then f(x) is into.<br \/>\nDiagrammatically into function can be shown as<\/p>\n<p>Note: If a function is onto, it cannot be into and vice versa. A polynomial of degree even will always be into. Thus a function can be one of these four types:<\/p>\n<ul>\n<li>one\u2212one onto (injective &amp; surjective)<\/li>\n<li>one\u2212one into (injective but not surjective)<\/li>\n<li>many\u2212one onto (surjective but not injective)<\/li>\n<li>many\u2212one into (neither surjective nor injective)<\/li>\n<\/ul>\n<p>Note:<strong><br \/>\n<\/strong>(i) If f is both injective &amp; surjective, then it is called a Bijective mapping. The bijective functions are also named as invertible, non singular or biuniform functions.<br \/>\n(ii) If a set A contains n distinct elements then the number of different functions defined from A\u2192 A is nn &amp; out of it n ! are one one.<br \/>\n<strong>Identity function:<\/strong> The function f: A \u2192 A defined by f(x) = x \u2200 x \u2208 A is called the identity of A and is denoted by IA. It is easy to observe that identity function is a bijection.<br \/>\n<strong>Constant function:<\/strong> A function f: A \u2192 B is said to be a constant function if every element ofA has the same f image in B. Thus f: A \u2192 B; f(x) = c, \u2200 x \u2208 A, c \u2208 B is a constant function. Note that the range of a constant function is a singleton and a constant function may be one-one or many-one, onto or into.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Algebraic_Operations_On_Functions\"><\/span>Algebraic Operations On Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If f &amp; g are real valued functions of x with domain set A, B respectively, then both f &amp; g are defined in A \u2229 B. Now we define f+g, f\u2212g, (f. g) &amp; (f\/g) as follows:<br \/>\n\\(\\left.\\begin{array}{ll}{\\text { (i) }} &amp; {(\\mathrm{f} \\pm \\mathrm{g})(\\mathrm{x})=\\mathrm{f}(\\mathrm{x}) \\pm \\mathrm{g}(\\mathrm{x})} \\\\ {\\text { (ii) }} &amp; {(\\mathrm{f} . \\mathrm{g})(\\mathrm{x})=\\mathrm{f}(\\mathrm{x}) \\cdot \\mathrm{g}(\\mathrm{x})}\\end{array}\\right] \\text { domain in each case is } \\mathrm{A} \\cap \\mathrm{B}\\)<br \/>\n(iii) \\(\\left(\\frac{f}{g}\\right)(x) = \\frac {f(x)}{g(x)}\\) domain is { x | x \u2208 A \u2229 B s . t g(x) \u2260 0}.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Composite_Of_Uniformly_Non-Uniformly_Defined_Functions\"><\/span>Composite Of Uniformly &amp; Non-Uniformly Defined Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Let f : A \u2192 B &amp; g : B \u2192 C be two functions . Then the function gof : A \u2192 C defined by (gof) (x) = g (f(x)) \u2200 x \u2208 A is called the composite of the two functions f &amp; g .<br \/>\nDiagrammatically \\(\\stackrel{x}{\\longrightarrow}[f] \\stackrel{f(x)}{\\longrightarrow}[g] \\longrightarrow g(f(x))\\). Thus the image of every x \u2208 A under the function gof is the g\u2212image of the f\u2212image of x.<br \/>\nNote that gof is defined only if \u2200 x \u2208 A, f(x) is an element of the domain of g so that we can take its g-image. Hence for the product gof of two functions f &amp; g, the range of f must be a subset of the domain of g.<br \/>\n<strong>Properties Of Composite Functions:<\/strong><br \/>\n(i) The composite of functions is not commutative i.e. gof \u2260 fog .<br \/>\n(ii) The composite of functions is associative i.e. if f, g, h are three functions such that fo (goh) &amp; (fog) oh are defined, then fo (goh) = (fog) oh .<br \/>\n(iii) The composite of two bijections is a bijection i.e. if f &amp; g are two bijections such that gof is defined, then gof is also a bijection.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Homogeneous_Functions_Relations_and_Functions\"><\/span>Homogeneous Functions | Relations and Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A function is said to be homogeneous with respect to any set of variables when each of its terms is of the same degree with respect to those variables.<br \/>\nFor example 5x<sup>2<\/sup> + 3y<sup>2<\/sup> \u2212 xy is homogeneous in x &amp; y . Symbolically if, f(tx , ty) = t<sup>n<\/sup> .f(x , y) then f(x , y) is homogeneous function of degree n.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Bounded_Function_Relations_and_Functions\"><\/span>Bounded Function | Relations and Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A function is said to be bounded if |f(x)| \u2264 M , where M is a finite quantity.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Implicit_Explicit_Function_Relations_and_Functions\"><\/span>Implicit &amp; Explicit Function | Relations and Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A function defined by an equation not solved for the dependent variable is called an Implicit Function. For eg. the equation x<sup>3<\/sup>+y<sup>3<\/sup> = 1 defines y as an implicit function. If y has been expressed in terms of x alone then it is called an Explicit Function.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Inverse_of_A_Function_Relations_and_Functions\"><\/span>Inverse of A Function | Relations and Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Let f : A \u2192 B be a one\u2212one &amp; onto function, then their exists a unique function g : B \u2192 A such that f(x) = y \u21d4 g(y) = x, \u2200 x \u2208 A &amp; y \u2208 B . Then g is said to be inverse of f. Thus g = f<sup>-1<\/sup>: B \u2192 A = { (f(x), x) | (x, f(x)) \u2208 f}.<\/p>\n<p><strong>Properties of Inverse Function<br \/>\n<\/strong><\/p>\n<ul>\n<li>The inverse of a bijection is unique.<\/li>\n<li>If f : A \u2192 B is a bijection &amp; g : B \u2192 A is the inverse of f, then fog = IB and gof = I<sub>A <\/sub>, where I<sub>A<\/sub> &amp; I<sub>B<\/sub> are identity functions on the sets A &amp; B respectively.<br \/>\nNote that the graphs of f &amp; g are the mirror images of each other in the line y = x . As shown in the figure given below a point (x \u2018,y \u2018) corresponding to y = x<sup>2<\/sup> (x &gt;0) changes to (y \u2018,x \u2018 ) corresponding to y = \\(\\pm\\sqrt {x}\\), the changed form of x = \\(\\sqrt {y}\\).<\/li>\n<li>The inverse of a bijection is also a bijection .<\/li>\n<li>If f &amp; g two bijections f : A \u2192 B , g : B \u2192 C then the inverse of gof exists and (gof)\u22121 = f\u22121o g\u22121<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Odd_and_Even_Functions_Relations_and_Functions\"><\/span>Odd and Even Functions | Relations and Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If f (\u2212x) = f (x) for all x in the domain of \u2018f\u2019 then f is said to be an even function. e.g. f (x) = cos x ; g (x) = x\u00b2 + 3 . If f (\u2212x) = \u2212f (x) for all x in the domain of\u2018f\u2019 then fis said to be an odd function. e.g. f (x) = sin x ; g (x) = x<sup>3<\/sup> + x .<\/p>\n<p>Note:<\/p>\n<ul>\n<li>f (x) \u2212 f (\u2212x) = 0 =&gt; f (x) is even &amp; f (x) + f (\u2212x) = 0 =&gt; f (x) is odd.<\/li>\n<li>A function may neither be odd nor even.<\/li>\n<li>Inverse of an even function is not defined.<\/li>\n<li>Every even function is symmetric about the y\u2212axis &amp; every odd function is symmetric about the origin.<\/li>\n<li>Every function can be expressed as the sum of an even &amp; an odd function. e.g.<\/li>\n<li>only function which is defined on the entire number line &amp; is even and odd at the same time is f(x)= 0.<\/li>\n<li>If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd then f.g will be odd.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Periodic_Function_Relations_and_Functions\"><\/span>Periodic Function | Relations and Functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A function f(x) is called periodic if there exists a positive number T (T &gt; 0) called the period of the function such that f(x + T) = f(x), for all values of x with in the domain of x e.g. The function sin x &amp; cos x both are periodic over 2\u03c0 &amp; tan x is periodic over \u03c0<br \/>\nNote:<\/p>\n<ul>\n<li>f (T) = f (0) = f (\u2212T) , where \u2018T\u2019 is the period.<\/li>\n<li>Inverse of a periodic function does not exist.<\/li>\n<li>Every constant function is always periodic, with no fundamental period.<\/li>\n<li>If f(x) has a period T &amp; g (x) also has a period T then it does not mean that f(x) + g (x) must have a period T. e.g. f(x) = |sinx| + |cosx|.<\/li>\n<li>If f(x) has a period p, then \\(\\frac {1}{f(x)}\\text {and} \\sqrt {f(x)}\\) also has a period p.<\/li>\n<li>If f(x) has a period T then f(ax+b) has a period T\/a (a&gt;0).<\/li>\n<\/ul>\n<p><strong>General: <\/strong>If x, y are independent variables, then:<\/p>\n<ul>\n<li>f (xy) = f(x) + f(y) \u21d2 f(x) = k <i>l<\/i>n x or f(x) = 0 .<\/li>\n<li>f(xy) = f(x) . f(y) \u21d2 f(x) = x<sup>n<\/sup> , n \u2208 R<\/li>\n<li>f(x + y) = f(x) . f(y) \u21d2 f(x) = a<sup>kx<\/sup>.<\/li>\n<li>f(x + y) = f(x) + f(y) \u21d2 f(x) = kx, where k is a constant.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Relations and Functions If to every value of x belonging to some set E there corresponds one or several values [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Relations and Functions class 11","_yoast_wpseo_title":"%%title%%, Important types, Classification, Properties","_yoast_wpseo_metadesc":"Learn Relations and Functions topic of Maths in detail explained by subject experts, Register free for online tutoring 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