Functions - Domain, Range

Functions

Graphs of some simple polynomial functions
1) Constant function \( f:R \to R,\,\,\,i.e\,\,\,y = f(x) = k(K \in R) \)

y = K is a line parallel to x - axis

2) Identity function :  \( f:R \to R,\,\,y = f(x) = x \) 
 (i.e out put is same as in put i.e image = pre - image)  
  y = x is a line passing through origin and     
  also x and y co - ordinates are equal


Some symmetric figures :
    i) \( f:R \to R,f(x) = x^2 \)
Here the range the is the set of all  the non-negative real numbers i.e.  
Note : 
     graphof f(x) is symmetric about y - axis  

Rational functions :  Let  and f(x) is obtanied when p(x) is divided by Q(x)\( ( \ne 0) \)  i.e \( f(x) = \frac{{p(x)}} {{Q(x)}} \) then f(x) is called a rational function
Ex:(i) \( f(x) = \frac{{P(x)}} {{Q(x)}} = \frac{{x^2 - 3x + 2}} {{x^2 + 5x + 7}} \)
(ii)\( f(x) = \frac{{P(x)}} {{Q(x)}} = \frac{1} {{x^n }}\left( {x \ne 0,\,\,\,\,n \in N} \right) \)

Piece wise defined functions 
i) Modulus function :
\( f \to R \to R^ + \) such that y =f(x) =|x| =\( \left\{ \begin{gathered} x,\,\,\,\,\,\,\,x > 0 \hfill \\ - x,\,\,\,\,x < 0 \hfill \\ \,\,0,\,\,\,\,\,x = 0 \hfill \\ \end{gathered} \right. \)


ii) Signum function : 
 \( f(x) = sgn(x) = \left\{ \begin{gathered} \frac{{|x|}} {x}(or)\frac{x} {{|x|}}\,(x \ne 0) \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x \ne 0) \hfill \\ \end{gathered} \right. \) 
=\( \left\{ \begin{gathered} + 1,\,\,\,\,\,if\,\,\,\,\,x > 0 \hfill \\ - 1,\,\,\,\,if\,\,\,\,\,x < 0 \hfill \\ 0\,\,\,\,\,\,if\,\,\,\,\,\,x = 0 \hfill \\ \end{gathered} \right. \)


iii)     Greatest Integer function :
    \( f = R \to z \) such that f(x) =[x] = x, 
    if \( n \leqslant x < n + 1(n \in z) \), [x] in dicates the integer part of x, which is nearest and smaller integer to x. 
    It is also known as floor of x

EX : [2.8]=2, [0.78]=0, [5.0]=5 [-8.0725]=-9

Fractional part function :
    \( f:z \to [0,1] \) such that f(x) ={x} =f if n = I +f 
    where ‘f’ is fractional part and            
    I is integral part \( \left( {0 \leqslant f < 1} \right) \)


Transcendental functions:
All functions which are not algebraic functions are called  “TRANSCENDENTAL”  funcltions. 
Note : About trigonomatric functions, inverse trigonometric functions and their graphs, domain and range are explained in the chapter’Periodicity and graphs of trigonometry.

Logarithmic function
If \( f(x) = \log _a^x \,\,\,\,\left( \begin{gathered} a > 0 \hfill \\ a \ne 1 \hfill \\ \end{gathered} \right) \) and x > 0 is called a logarithmic function


Exponential Function :
If a > 0 , then f(x) = ax is called exponential function.
    a) y =f(x) = ax(01)
 
Domain of \( f(x) = a^x \) is \( \left( { - \infty ,\infty } \right) \) and 
    range of  f(x) = a^x is \( \left( {0,\infty } \right) \) (a > 0)