Addition of two real functions Let f : X → R and g : X → R be any two real functions , where X ⊂ R . Then , we define ( f + g ): X → R by ( f + g ) ( x ) = f ( x ) + g ( x ), for all x ∈ X .
Subtraction of a real function from another Let f : X → R and g : X → R be any two real functions , where X ⊂R . Then , we define ( f – g ) : X→R by ( f – g ) ( x ) = f ( x ) – g ( x ), for all x ∈ X .
Multiplication by a scalar Let f : X→R be a real valued function and α be a scalar . Here by scalar , we mean a real number . Then the product α f is a function from X to R defined by ( α f ) ( x ) = α f ( x ), x ∈X .
Multiplication of two real functions The product ( or multiplication ) of two real functions f : X→R and g : X→R is a function fg : X→R defined by ( fg ) ( x ) = f ( x ) g ( x ), for all x ∈ X . This is also called point wise multiplication .
( v ) Quotient of two real functions Let f and g be two real functions defined from X→R where X ⊂R . The quotient of f by g denoted by f / g is a
function defined by ,
, provided g ( x ) ≠ 0 , x ∈ X
Numerical : Let f ( x ) = x 2 and g ( x ) = 2x + 1 be two real functions . Find ( f + g ) ( x ), ( f - g ) ( x ), ( f g ) ( x ) & ( f / g ) ( x )
Solution : ( f + g ) ( x ) = x 2 + 2x + 1 , ( f – g ) ( x ) = x 2 – 2x – 1 , ( fg ) ( x ) = x 2 ( 2x + 1 ) = 2x 3 + x 2 , ( f / g ) ( x ) = x 2 / ( 2x + 1 )