Class 11 Maths Relation Functions Algebra of function

Algebra of function

• Addition of two real functions : f(x) + g(x)
• Subtraction of a real function from another : f(x)- g(x)
• Multiplication by a scalar : a* f(x)
• Multiplication of two real functions : f(x) * g(x)
• Quotient of two real functions : f(x) / g(x) where g(x)= 0

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→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) =x2and 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) = x2 + 2x + 1,

(f –g) (x) = x2 – 2x – 1,

(fg) (x) = x2 (2x + 1) = 2x3 + x2,

(f /g) (x) = x2 / (2x+ 1)

.