Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Some real functions and their graphs

(i) Constant Function:

A function f:A\rightarrow R is said to be a constant function if there is a real number  K such that  f(x) =K, \forall x \in A .

Hence \text {dom}f= A\subseteq R, \text {rng}f =\{K\} which is singleton.

Example:

f(x) =3 , x \in R , the line parallel to  x-axis .

(ii) Identify function:

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

(c) If a>1, a^x >a^y \text { iff } x >y

(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

(c) \log_ax=0 \Leftrightarrow x=1

(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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