Algebraic Operations on Functions

Here you will learn algebraic operations on functions, equal or identical functions and homogeneous function.

Let’s begin –

Algebraic Operations on Functions

Various operations, namely addition, subtraction, multiplication, division etc on real function are :

(i) Addition : Let f : $$D_1$$ $$\rightarrow$$ R and g : $$D_2$$ $$\rightarrow$$ R be two real functions. Then, their sum f + g is defined from $$D_1 \cap D_2$$ to R which associates each x $$\in$$ $$D_1 \cap D_2$$ to the number f(x) + g(x). It is defined as

(f + g) (x) = f(x) + g(x)  for all  x $$\in$$ $$D_1 \cap D_2$$

(ii) Difference (Subtraction) : Let f : $$D_1$$ $$\rightarrow$$ R and g : $$D_2$$ $$\rightarrow$$ R be two real functions. Then, the difference of g from f is denoted by f – g and is defined as

(f – g) (x) = f(x) – g(x)  for all  x $$\in$$ $$D_1 \cap D_2$$

(iii) Product : Let f : $$D_1$$ $$\rightarrow$$ R and g : $$D_2$$ $$\rightarrow$$ R be two real functions. Then, their product (or pointwise multiplication) f g is a function $$D_1 \cap D_2$$ to R and is defined as

(f g) (x) = f(x) g(x)  for all  x $$\in$$ $$D_1 \cap D_2$$

(iv) Quotient : Let f : $$D_1$$ $$\rightarrow$$ R and g : $$D_2$$ $$\rightarrow$$ R be two real functions. Then, the quotient of f by g is denoted by $$f\over g$$ and it is a function from $$D_1 \cap D_2$$ – {x : g(x) = 0} to R defined by

($$f\over x$$)(x) = $$f(x)\over g(x)$$ for all  x $$\in$$ $$D_1 \cap D_2$$ – {x : g(x) = 0}

(v) Multiplication of a function by a scalar : Let f : D $$\rightarrow$$ R be a real function and $$\alpha$$ be a scalar (real number). Then the product $$\alpha$$f is a function from D to R and is defined as

$$\alpha f$$ (x) = $$\alpha$$ f(x) for all x $$\in$$ D

Example : Find the domain of the function y = $$log_{(x-4)}(x^2 – 11x + 24)$$.

Solution : Here ‘y’ would assume real value if,

x – 4 > 0 and $$\ne$$ 1, $$x^2 -11x + 24$$ > 0 $$\implies$$ x > 4 and $$\ne$$ 5, (x-3)(x-8) > 0

$$\implies$$ x > 4 and $$\ne$$ 5, x < 3 or x > 8 $$\implies$$ x > 8 $$\implies$$ Domain (y) = (8, $$\infty$$)

Bounded Function

A function is said to be bounded if there exists a finite M such that |f(x)| $$\le$$ M, $$\forall$$ x $$\in$$ $$D_f$$.