JEE Main & Advanced

If \[f:A\to B\] be a one-one onto (bijection) function, then the mapping \[{{f}^{-1}}:B\to A\] which associates each element \[b\in B\] with element \[a\in A,\] such that \[f(a)=b,\] is called the inverse function of the function \[f:A\to B\].     \[{{f}^{-1}}:B\to A,\,\,{{f}^{-1}}(b)=a\Rightarrow f(a)=b\]     In terms of ordered pairs inverse function is defined as \[{{f}^{-1}}=(b,\,a)\] if \[(a,\,\,b)\in f\].     For the existence of inverse function, it should be one-one and onto.     Properties of Inverse function :     (1) Inverse of a bijection is also a bijection function.     (2) Inverse of a bijection is unique.     (3) \[{{({{f}^{-1}})}^{-1}}=f\]     (4) If \[f\] and \[g\] are two bijections such that \[(gof)\] exists then \[{{(gof)}^{-1}}={{f}^{-1}}o{{g}^{-1}}\].     (5) If \[f:A\to B\] is a bijection then \[{{f}^{-1}}\,.\,B\to A\] is an inverse function of \[f.\,{{f}^{-1}}\] \[of={{l}_{A}}\] and \[fo{{f}^{-1}}={{l}_{B}}\]. Here \[{{l}_{A}},\] is an identity function on set A, more...

• If \[f:A\to B\] and \[g:B\to C\] are two function then the composite function of \[f\] and \[g,\]

    \[gof\,A\to C\] will be defined as \[gof(x)=g\,[f(x)],\,\forall x\in A\]     (1) Properties of composition of function :     (i) \[f\] is even, \[g\] is even \[\Rightarrow \]\[fog\] even function.      (ii) \[f\] is odd, \[g\] is odd \[\Rightarrow \]\[fog\] is odd function.     (iii) \[f\] is even, \[g\] is odd   \[\Rightarrow \]\[fog\] is even function.                (iv) \[f\] is odd, \[g\] is even \[\Rightarrow \]\[fog\] is even function.                  (v) Composite of functions is not commutative i.e., \[fog\,\ne \,gof\].     (vi) Composite of functions is associative i.e., \[(fog)oh\,=\,fo(goh)\]                    (vii) If \[f:A\to B\] is bijection and \[g:B\to A\] is inverse of  \[f\]. Then \[fog={{I}_{B}}\] and \[gof={{I}_{A}}.\]     where, \[{{I}_{A}}\] and \[{{I}_{B}}\] are identity functions on the sets A more...

A function is said to be periodic function if its each value is repeated after a definite interval. So a function \[f(x)\] will be periodic if a positive real number \[T\] exist such that, \[f(x+T)=f(x)\], \[\forall x\in \]domain. Here the least positive value of \[T\] is called the period of the function.  

(1) Even function : If we put \[(-x)\] in place of \[x\] in the given function and if \[f(-x)=f(x)\], \[\forall x\in \] domain then function \[f(x)\] is called even function. e.g. \[f(x)={{e}^{x}}+{{e}^{-x}},\] \[\,f(x)={{x}^{2}},\,\] \[f(x)=x\sin x,\,\]\[\,f(x)=\cos x,\,f(x)={{x}^{2}}\cos x\] all are even functions.     (2) Odd function : If we put \[(-x)\] in place of \[x\] in the given function and if \[f(-x)=-f(x),\,\,\forall x\in \] domain then \[f(x)\] is called odd function. e.g., \[f(x)={{e}^{x}}-{{e}^{-x}}\], \[f(x)=\sin x,\,f(x)={{x}^{3}}\], \[f(x)=x\cos x,\] \[f(x)={{x}^{2}}\sin x\] all are odd functions.     Properties of even and odd function    

• The graph of even function is always symmetric with respect to y-axis. The graph of odd function is always symmetric with respect to origin.

 

• The product of two even functions is an even function.

 

• The sum and difference of two even functions is an more...

(1) One-one function (injection) : A function \[f:A\to B\] is said to be a one-one function or an injection, if different elements of A have different images in B. Thus, \[f:A\to B\] is one-one.     \[a\ne b\,\,\Rightarrow \,\,f(a)\ne f(b)\] for all \[a,\,\,b\in A\]      \[\Leftrightarrow \,\,f(a)=f(b)\,\,\Rightarrow \,\,a=b\] for all \[a,\,\,b\in A\].     e.g. Let \[f:A\to B\] and \[g:X\to Y\] be two functions represented by the following diagrams.         Clearly, \[f:A\to B\] is a one-one function. But \[g:X\to Y\] is not one-one function because two distinct elements \[{{x}_{1}}\] and \[{{x}_{3}}\] have the same image under function \[g\].     (i) Method to check the injectivity of a function     Step I : Take two arbitrary elements \[x,\,\,y\] (say) in the domain of \[f\].     Step II : Put \[f(x)=f(y).\] more...

(1) Scalar multiplication of a function : \[(c\,f)(x)=c\,f(x),\]  where \[c\] is a scalar. The new function \[c\,f(x)\] has the domain \[{{X}_{f}}.\]     (2) Addition/subtraction of functions     \[(f\pm g)(x)=f(x)\pm g(x).\] The new function has the domain \[X\].     (3) Multiplication of functions     \[(fg)(x)=(g\,f)(x)=f(x)g\,(x).\] The product function has the domain \[X\].     (4) Division of functions :     (i) \[\left( \frac{f}{g} \right)\,(x)=\frac{f(x)}{g(x)}.\] The new function has the domain \[X,\] except for the values of \[x\] for which \[g\,(x)=0.\]     (ii) \[\left( \frac{g}{f} \right)\,(x)=\frac{g(x)}{f(x)}.\] The new function has the domain \[X,\] except for the values of \[x\] for which \[f(x)=0.\]     (5) Equal functions : Two function \[f\] and \[g\] are said to be equal functions, if and only if     (i) Domain of \[f=\] Domain of \[g\]       (ii)  Co-domain of \[f=\] Co-domain of \[g\]      (iii) \[f(x)=g(x)\,\forall x\in more...

If a function \[f\] is defined from a set \[f\] to set B then for \[f:A\to B\] set A is called the domain of function \[f\] and set \[B\] is called the co-domain of function \[f\]. The set of all f-images of the elements of \[A\] is called the range of function \[f\].     In other words, we can say      Domain = All possible values of \[x\] for which \[f(x)\] exists.               Range   = For all values of \[x,\] all possible values of \[f(x)\].               (1) Methods for finding domain and range of function   (i) Domain   (a) Expression under even root (i.e., square root, fourth more...

(1) Function can be easily defined with the help of the concept of mapping. Let \[X\] and \[Y\] be any two non-empty sets. “A 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 \[Y''\]. Let the correspondence be \['f'\] then mathematically we write \[f:X\to Y\] where \[y=f(x),\,x\in X\] and \[y\in Y.\] We say that \['y'\] is the image of \['x'\] under \[f\] (or \[x\] is the pre image of \[y\]).     Two things should always be kept in mind:     (i) A mapping \[f:X\to Y\] is 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 \[X\].   (ii) Every element in more...

  There are four types of interval:     (1) Open interval : Let a and b be two real numbers such that \[a<b\], then the set of all real numbers lying strictly between \[a\] and \[b\] is called an open interval and is denoted by \[[a,\,\,b]\] or \[(a,\,\,b)\]. Thus, \[[a,\,\,b]\] or  \[(a,\,\,b)=\{x\in R\,:\,a<x<b\}\].       (2) Closed interval : Let a and b be two real numbers such that \[a<b,\] then the set of all real numbers lying between \[a\] and \[b\] including \[a\] and \[b\] is called a closed interval and is denoted by \[[a,\,\,b]\]. Thus, \[[a,\,\,b]=\{x\in R\,:\,a\le x\le b\}\]     (3) Open-Closed interval : It is denoted by \[[a,\,\,b]\] or \[(a,\,\,b]\] and \[[a,\,\,b]\] or \[(a,\,\,b]=\{x\in R\,:\,\,a<x\le b\}\].     (4) Closed-Open more...

(1) Real numbers : Real numbers are those which are either rational or irrational. The set of real numbers is denoted by \[R\].  

• (2) Related quantities : When two quantities are such that the change in one is accompanied by the change in other, e., if the value of one quantity depends upon the other, then they are called related quantities.

    (3) Variable: A variable is a symbol which can assume any value out of a given set of values.     (i) Independent variable : A variable which can take any arbitrary value, is called independent variable.     (ii) Dependent variable : A variable whose value depends upon the independent variable is called dependent variable.     (4) Constant : A constant is a symbol which does not change its value, i.e., retains the same value throughout a set of mathematical more...