Algebra of Functions

A set of all rational and irrational numbers is called real numbers, and it is denoted by R.

A relation f from a non-empty set A to a non-empty set B is said to be a function, if:
(i) an element of set A is associated to a unique element in set B.
(ii) all the elements of set A are associated to the elements of set B.

The set of all the real numbers for which a real valued function is defined is called the domain of that function.
If f and g are two real valued functions with domains A and B respectively then their sum, difference, product and quotient are defined as follows:

(f + g)(x) = f(x) + g(x), for all x ? A ? B
(f – g)(x) = f(x) – g(x), for all x ? A ? B
(f.g)(x) = f(x)g(x), for all x ? A ? B
(f/g)(x) =f(x)/g(x), provided g(x) ? 0, x ? A ? B

The product (? f) is a real function and defined by (? f)(x) = ? f(x), x ? A.
Where, f be a real function with domain A and ? be a scalar.
Keywords: Multiplication by a Scalar.

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