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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