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 \(\in\) \(D_f\).
