Functions - Domain, Range

Functions

Typess of Functions

Algebraic Functions :
    Functions consisting of finite number of terms of different powers of independent variable (x ) and the operations \( + ,\,\, - ,\,\, \times \,\,and\,\, \div \) are called algebraic functions 
Ex : \( 6x^2 + x^{1/3} - 1,\,\,\frac{{x^2 + x - 1}} {{x^2 + 3x - 2}} \) and etc
Polynomial function : 
    Let \( f:R \to R \) be a polynomial function such that  
\( f(x) = a_0 + a_1 x + a_2 x^2 + .......a_n x^n \,\,(n \in N) \) is called polynomial function 
    a₁, a₂,......a_n are real numbers 
    Ex : \( 3x^4 + 6x^3 - \frac{1} {2}x^2 + 1,\,\,\,\sqrt 2 x^6 - \sqrt 5 x^4 + \sqrt 2 x + \frac{{\sqrt 3 }} {2} \)
Note :  Let \( f:A \to B \), if A and B subsets of R, then 'f' is called real function

Operations on real functions : 
    Set f and g are two real functions with domains D₁ and D₂ respectively. The some composite functions like \( f \pm g,\,\,fg\,and\frac{f} {g} \) are defined on the domain \( D_1 \cap D_2 \)
1)\( f + g:D_1 \cap D_2 \to R,\,\,\,\,\,\forall x \in D_1 \cap D_2 \)
2)\( f - g:D_1 \cap D_2 \to R,\,\,\,\,\,\forall x \in D_1 \cap D_2 \)
3)\( fg:D_1 \cap D_2 \to R,\,\,\,\,\,\forall x \in D_1 \cap D_2 \)
4)\( \frac{f} {g}:\left( {D_1 \cap D_2 } \right) - \left\{ {x/g(x) \ne 0} \right\} \to R\,\,\forall x \in D_1 \cap D_2 - \left\{ {x/g(x) \ne 0} \right\} \)

Ex :    If , g={(4,-4),(6,5),(8,5)} then the find f+g, f-g, fg and f/g
SOL : D₁ = {4,5,6}, D₂={4,6,8}
    \( \therefore D_1 \cap D_2 = \{ 4,6\} \)
    i) f +g =\( \{ (4,5 + ( - 4),(6, - 4 + 5)\} = \{ (4,1),(6,1)\} \)
    ii) f-g ={(4,5-(-4), (6,-4-5)} ={(4,9),(6, -9)}
    iii) fg = {(4,(5)(-4), (6, (-4) (+5)} = {(4,-20), (6, -20)}
    iv) \( \frac{f} {g} = \left\{ {\left( {4,\frac{{ - 5}} {4}} \right),\left( {6,\frac{{ - 4}} {5}} \right)} \right\} \)