For any non-empty set $A\subseteq R$ , the function $f: A\rightarrow A$ is defined by $f(x)=x, \forall x\in A$ is called the identity function on $A$ . It is denoted by $Id_A$ .

Hence $\text { dom }f =rng f$

$\rightarrow$ The line plotted through origin.

(iii) Polynomial Function:

A function $f:A\rightarrow R$ defined by $f(x)=a_0+a_1 x+a_2x^2+…+a_nx^n$ , where $n$ is a non-negative integer and $a_0,a_1,…,a_n$ are real constant with $a_n\neq 0$ , is called a polynomial function or a polynomial of degree $n$ .

Example:

$f(x)=x^2-1,\forall x \in R$

$x$	$0$	$1$	$-1$
$f(x)$	$-1$	$0$	$0$

(iv) Rational Function:

A function $f(x)=\dfrac{p(x)} {q(x)}$ , where $p(x)$ & $q(x)$ are polynomials with $q(x)\neq 0, \forall x \in \text{dom }f$ , is called a rational function.

Example:

$f(x)= \dfrac{x-1}{x+2}$ , $x+1\neq 0 \implies x\neq -1$

$x$	$0$	$1$	$2$	$-2$
$y=f(x)$	$-1$	$0$	$\dfrac{1}{3}$	$3$

(v) Modulus function:

If $f: R \rightarrow R$ is defined by

$f(x) = | x | = \begin{Bmatrix}x, & x \geq 0\\ -x,& x < 0\end{Bmatrix}$

is called Modulus function.

$\rightarrow$ The modulus function is also known as absolute value function.

$\rightarrow$ Its domain $R$ and rage is

$R^+ \cup \{0\}= \{x \in R | x \geq 0\}$

Example:

$y =| x |$

(vi) Signum function:

The signum function on $R$ is defined by

$\text {Sgnx }=\begin{Bmatrix}\dfrac{1}{1x1}, & (x \neq 0) \\ 0,& (x=0)\end{Bmatrix}$

The range of $\text { Sgnx }$ is $\{-1,0,1\}$ .

(vii) Exponential function:

An exponential function is defined by

$f(x)=a^x (a>0, a\neq 1), x \in R$ .

The fact that $a^x$ exists for every $x \in R$ .

Example:

$y=2^x$

(a) $a^{x+y}=a^xa^y, a>0, (a^x)^y=a^{xy},x,y \in R$

(b) $a^x =1 \text { iff } x=0$

(d) If $a<1$ then $a^x < a^y \text { iff } x>y$

(e) $a^x$ is closer to the $x$ -axis as $x$ recedes away from zero along negative values.

$\text { domf}= R$ and $\text { rngf}= R^+$

(viii) Logarithmic function:

The function $f$ defined by $f(x)= \log_a x, (a>0,a\neq 1)$ where $y= \log_ax\Leftrightarrow a^y=x$ is called the logarithmic function.

The graphs meets the $x$ -axis at $(1,0)$ and never meet the $y$ – axis.

Some Logarithmic function:

(a) $\log_a(xy)=\log_ax+\log_ay$

(b) $\log_a \left ( \dfrac{x}{y} \right )= \log_a x -\log_a y$

(d) $\log_xx=1$

(e) $\log_ax=\dfrac{1}{\log _x a} , x\neq 1$

(f) $\log _ax =\log_bx \cdot\log_a b$

(g) If $a>1 , \log_ax > \log_ay \text { iff } x>y$ and if $a < 1, \log_ax<\log_ay \text { iff } x>y$

(h) $\dfrac{\log_ax}{\log_ay}=\log_yx, (y\neq 1)$

(ix) Greatest Integer Function:

The function $f$ is defined by

$f(x): [x]$

where $[x]$ is the greatest integer not greater that $x$ (less than or equal to $x$ ) is called the greatest integer function.

$[x]= n \text { for } n \leq x <n+1$

(a) $\text {domf }= R (b) \text { rngf } =Z$

$f(x)= [x]=\begin{Bmatrix}0 & \text {if} & 0 \leq x < 1 \\ 1 & \text {if} &1\leq x < 2 \\ 2 & \text {if} & 2 \leq x < 3...\end{Bmatrix}$

and

$\begin{Bmatrix}-1 &\text {if}& -1 \leq 0<x \\ -2& \text {if}& -2\leq x <-1\\ -3&\text {if}&-3\geq x < -2...\end{Bmatrix}$

The graph consists of infinitively many closed open parallel line segments.

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