Functions & Relations - Real Functions & Algebra of functions

Functions are one of the very important concepts in mathematics which have got numerous applications in the real world. Be it the mega skyscrapers or the superfast cars, their modeling requires methodical application of functions. Almost all the real world problems are formulated interpreted and solved using  real functions.

A relation is a rule that “relates” an element from one set to an element from another set. A function is a special kind of relation. A relation F is said to be a

function if each element in set A is associated with exactly one element in set B. For example a relation F from set A to set B such that it associates a natural number to its square is a function because for every element in set A, we will have exactly one association in set B.

Real Functions

Now that we understand what functions are let us discuss how we perform mathematical operations on real functions such as addition of two or more  functions, subtraction of two functions, multiplying a real function by a real number etc.

  • Adding two real functions

For adding two real functions let us define the functions f and g such that f: X ⟶R and g: X ⟶R are two real functions such that X is a subset of R. Then (f + g): X ⟶R can be defined as:

 

(f + g)(x)=f(x) + g(x), for all x ϵ X

  • Subtracting two functions

For subtracting two real functions let us define the functions f and g such that f: X ⟶R and g: X ⟶R are two real functions such that X is a subset of R. Then (f – g): X ⟶R can be defined as:

 

(f – g)(x)=f(x) – g(x), for all x ϵ X

  • Multiplying a real function by a scalar (real number)

Let us define a real function f such that f: X ⟶R, X⊆R and a ϒ be a real scalar quantity. Then the product of scalar ϒ and the function f is also a function defined from X to R as:

 

(ϒf)(x) = ϒf(x), for all x ϵ X

  • Multiplying two functions:

For multiplying two real functions let us define the functions f and g such that f: X ⟶R and g: X ⟶R are two real functions such that X is a subset of R. Then fg: X ⟶R can be defined as:

(fg)(x) = f(x)g(x), for all x ϵ X

 

  • Quotient of two functions:

For determining the quotient of two real functions let us define the functions f and g such that f: X ⟶R and g: X ⟶R are two real functions such that X is a subset of R. Then f/g: X ⟶R can be defined as:Real FunctionsGiven that g (x) ≠ 0, for all x ɛ X

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