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Algebra of Real Functions

description

  • Sum, Difference,Product and Quotient of Function
  • Addition of two real functions
  • Subtraction of a real function from another
  • Multiplication by a scalar
  • Multiplication of two real functions
  • Quotient of two real functions

notes

Here,  we shall learn how to add two real functions, subtract a real function from another, multiply a real function by a scalar (here by a scalar we mean a real number), multiply two real functions and divide one real function by another.

(i) 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.
Example- `f(x)= x^2+1 and g(x)= sqrt(x-1)`
`(f+g)(x)= x^2+1+ sqrt(x-1)`

(ii) 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. 
Example- `f(x)= x^2+1 and g(x)= sqrt(x-1)`
`(f-g)(x)= x^2+1- sqrt(x-1)`

(iii) 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. (kf)(x)= kf(x), where k is a constant.
Example 1- `f(x)= x^2+1 and g(x)= sqrt(x-1)`
`(fg)(x)= (x^2+1)[sqrt(x-1)]`

Example 2- `f(x)= x^2+1`
`3f(x)= 3(x^2+1)`

(iv) 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 is a function defined by, `(f/g)(x)=[f(x)]/[g(x)]`, provided g(x) ≠ 0, x ∈ X.

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