Algebra of Real Functions

 Let f : X → R and g : X → R be any two real functions, where X $\subset$ R.

(i) Addition of 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

(ii) Subtraction of two real 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

(iii) Multiplication by a scalar

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

       ($\alpha$ f)(x) = $\alpha$f (x), for all x ϵ X

(iv) Multiplication of two real 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:

       f(g(x)) = f(x) g(x), for all x ϵ X

(v) Quotient of two real 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:

      $\left(\frac{f}{g}\right)$(x) = $\frac{f(x)}{g(x)}$, g(x) $\neq$ 0

Given that g (x) ≠ 0, for all x ϵ X

Simple Problem: 

Let f(x) = x³ and g(x) = 3x + 1 and a scalar, $\alpha$ = 6. Find

1. (f + g) (x)
2. (f – g) (x)
3. ($\alpha$f) (x)
4. ($\alpha$g) (x)
5. (fg) (x)
6. ($\frac{f}{g}$) (x)

Answer : 

We have,

1. (f + g) (x) = f(x) + g(x) = x³ + 3x + 1.
2. (f – g) (x) = f(x) – g(x) = x³ – (3x + 1) = x³ – 3x – 1.
3. ($\alpha$f) (x) = $\alpha$ f(x) = 6x³ 
4. ($\alpha$g) (x) = $\alpha$ g(x) = 6 (3x + 1) = 18x + 6.
5. (fg) (x) = f(x) g(x) = x³ (3x +1) = 3x⁴ + x³.
6. ($\frac{f}{g}$) (x) = $\frac{f(x)}{g(x)}$ = $\frac{x^3}{3x+1}$, provided x ≠ –$\frac{1}{2}$.

Contents:

• Ordered Pair
• Cartesian Product
• Relations
• Different Types of Relations
• Composition of Relations
• Functions or Mappings
• Classification of Functions 
• Modulus Function
• Greatest Integer Function
• Inverse Function
• Algebra of Real Functions
• Composition of Functions