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Inverse Function

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, and ${{l}_{B}},$ is an identity function on set B.

Composite Function

• 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 and B respectively.     (viii) If $f:A\to B$ and $g:B\to C$ are two bijections, then $gof:A\to C$ is bijection and ${{(gof)}^{-1}}=({{f}^{-1}}o{{g}^{-1}}).$     (ix) $fog\ne gof$ but if, $fog=gof$ then either ${{f}^{-1}}=g$ or ${{g}^{-1}}=f$ also, $(fog)\,(x)=(gof)\,(x)=(x).$     (x) $gof(x)$ is simply the g-image of $f(x),$ where $f(x)$ is f-image of elements $x\in A$.     (xi) Function $gof$ will exist only when range of $f$ is the subset of domain of $g$.     (xii) $fog$ does not exist if range of g is not a subset of domain of $f$.     (xiii) $fog$ and $gof$ may not be always defined.     (xiv) If both $f$ and $g$ are one-one, then $fog$ and $gof$ are also one-one.     (xv) If both $f$ and $g$ are onto, then $gof$ is onto.

Periodic Function

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.

Even and Odd 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 even function.

• The sum and difference of two odd functions is an odd function.

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

• The product of an even and an odd function is an odd function. It is not essential that every function is even or odd. It is possible to have some functions which are neither even nor odd function. g.$f(x)={{x}^{2}}+{{y}^{3}},\,\,f(x)={{\log }_{e}}\,x,\,\,f(x)={{e}^{x}}$.

• The sum of even and odd function is neither even nor odd function.
Zero function $f(x)=0$ is the only function which is even and odd both.

Kinds of function

(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 $f(x)=f(y).$     Step III : Solve $f(x)=f(y).$ If $f(x)=f(y)$ gives $x=y$ only, then $f:A\to B$ is a one-one function (or an injection). Otherwise not.     If function is given in the form of ordered pairs and if two ordered pairs do not have same second element then function is one-one.     If the graph of the function $y=f(x)$ is given and each line parallel to x-axis cuts the given curve at maximum one point then function is one-one. e.g. (ii) Number of one-one functions (injections) : If $A$ and $B$ are finite sets having $m$ and $n$ elements respectively, then number of one-one functions from  $A$ to $B$=\left\{ \begin{align} & ^{n}{{P}_{m}},\,\,\,\text{if }n\ge m \\ & \,\,0\,\,\,\,,\,\,\,\text{if }n · If function is given in the form of set of ordered pairs and the second element of atleast two ordered pairs are same then function is many-one. · If the graph of \[y=f(x) is given and the line parallel to x-axis cuts the curve at more than one point then function is many-one. (3) Onto function (surjection) : A function $f:A\to B$ is onto if each element of B has its pre-image in A. Therefore, if ${{f}^{-1}}(y)\in A,\,\,\forall y\in B$ then function is onto. In other words, Range of $f=$ Co-domain of f. e.g. The following arrow-diagram shows onto function. Number of onto function (surjection) : If A and B are two sets having $m$ and $n$ elements respectively such that $1\le n\le m,$ then number of onto functions from A to B is  $\sum\limits_{r=1}^{n}{{{(-1)}^{n-r}}{{\,}^{n}}{{C}_{r}}{{r}^{m}}.}$     (4) Into function : A function $f:A\to B$ is an into function if there exists an element in B having no pre-image in A.     In more...

Algebra of Functions

(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) $f(x)=g(x)\,\forall x\in$ their common domain     (6) Real valued function : If $R,$ be the set of real numbers and $A,\,\,B$ are subsets of $R,$ then the function $f:A\to B$ is called a real function or real–valued function.

Domain, Co-domain and Range of Function

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 root etc.) $\ge 0$.  Denominator $\ne 0$.   If domain of $y=f(x)$ and $y=g\,(x)$ are ${{D}_{1}}$ and ${{D}_{2}}$ respectively then the domain of $f(x)\pm g\,(x)$ or $f(x).g\,(x)$ is ${{D}_{1}}\cap {{D}_{2}}.$   While domain of $\frac{f(x)}{g(x)}$ is ${{D}_{1}}\cap {{D}_{2}}-\{g(x)=0\}.$     Domain of $\left( \sqrt{f(x)} \right)={{D}_{1}}\cap \{x:f(x)\ge 0\}$     (ii) Range : Range of $y=f(x)$ is collection of all outputs $f(x)$ corresponding to each real number in the domain.     (a) If domain $\in$ finite number of points $\Rightarrow$ range $\in$ set of corresponding $f(x)$ values.     (b) If domain $\in R$ or $R-$ [some finite points]. Then express $x$ in terms of $y$. From this find $y$ for $x$ to be defined (i.e., find the values of $y$ for which $x$ exists).     (c) If domain $\in$ a finite interval, find the least and greatest value for range using monotonicity.

Definition of Function

(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 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. (2) Testing for a function by vertical line test : A relation $f:A\to B$ is 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. (3) Number of functions : 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 ${{n}^{m}}$.       (4) Value of the function : 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 $f(a)$.       e.g. If $f(x)={{x}^{2}}+1,$ then $f(1)={{1}^{2}}+1=2,$ $f(2)={{2}^{2}}+1=5,$ $f(0)={{0}^{2}}+1=1$ etc.

Intervals

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 interval : It is denoted by $[a,\,\,b]$ or $[a,\,\,b)$ and  $[a,\,\,b]$ or $[a,\,\,b)=\{x\in R\,:\,\,a\le x<b\}$ Some Important Definitions

(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 operation. These are generally denoted by $a,\,\,b,\,\,c$ etc. There are two types of constant, absolute constant and arbitrary constant.     (5) Absolute value : The absolute value of a number $x,$ denoted by $|x|,$ is a number that satisfies the conditions     $|x|=\left\{ \begin{matrix} -x \\ \,\,0 \\ \,\,\,x \\ \end{matrix}\,\,\,\begin{matrix} \text{if} \\ \text{if} \\ \text{if} \\ \end{matrix}\,\,\begin{matrix} x<0 \\ x=0 \\ x>0 \\ \end{matrix}. \right.$ We also define $|x|,$as follows, $|x|=$ maximum $\{x,\,\,-x\}$ or $|x|=\sqrt{{{x}^{2}}}$.     (6) Fractional part : We know that $x\ge [x].$ The difference between the number $'x'$ and it’s integral value $'[x]'$ is called the fractional part of $x$ and is symbolically denoted as $\{x\}$. Thus, $\{x\}=x-[x]$e.g., if $x=4.92$ then $[x]=4$ and $\{x\}=0.92$.     Fractional part of any number is always non-negative and less than one.

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