JEE Main & Advanced Mathematics Functions 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.