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)