Algebra of functions

We study addition, subtraction, multiplication and division in functions

Here the functions which we are taking are real functions

i.e. f: X → R

Where X is a subset of R and R is real numbers

Example

• f: R → R is a real function
• f: Z → R is a real function
  (Where Z is set of integers and integer is a subset of Real numbers i.e. Z ⊂ R )

Let f: R → R , g: R → R

f(x) = x, g(x) = x – 1

Addition

f + g = x + (x – 1) = 2x – 1

Subtraction

f – g = x – (x – 1) = x – x + 1 = 1

Multiplication by scalar

2f = 2 ×  f(x) = 2x

9g = 9 × g(x) = 9(x – 1) = 9x – 9

Multiplication

fg = x × (x – 1) = x ² – x

Division

f/g = x/(x - 1)

Where x ∈ R and x – 1 ≠ 0

i.e. x ≠ 1

So, x will be all numbers except 1

So, x ∈ R – {1}

So, for f: X → R and g: X → R

(f + g) (x) = f(x) + g(x), x ∈ X

(f – g) (x) = f(x) + g(x), x ∈ X

(kf) (x) = k f(x), x ∈ X, k is a real number

(fg) (x) = f(x) × g(x), x ∈ X

(f/g) (x) = f(x)/g(x), x ∈ X, and g(x) ≠ 0