# 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) = xand 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)

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