Just like the real numbers which can be divided, multiplied, added, and subtracted to generate other numbers, there’s a helpful way of dividing, multiplying, subtracting, and adding functions for producing different functions. And we describe these operations like the following:
Given functions f and g, their sum f + g, difference f – g, product f.g & quotient f/g are described by:
(f + g) (x) = f (x) + g (x)
(f – g) (x) = f (x) – g (x)
(f.g) (x) = f (x) . g (x)
(f/g) (x) = f (x) / g (x)
For the given functions like f. g, f + g, and f – g, the domain is illustrated as domain’s intersection of f and g, and for the function, f/g, the domain is defined as the intersection with points where the excluded function is g(x) = 0.
Illustration
Let f and g be the functions f(x) = √9-x & g(x) = √x-11
Accordingly, the formulas for f. g, f + g, and f – g
So, (f/g) (x) = f (x) / g (x)
Hence, √9-x / √x-11
Since f’s domain is (-∞, 9], whereas of g is [11, ∞), and the domain f.g, f – g, and f + g is the interval [9, 11]. Since the value of function, g(x) = 0 when x = 9, we should be excluding this point for seeing the domain of f/g as (9, 11].
Assuming the functions f and g in the form f(x) = sin x and g(x) = √4-cosx, accordingly the formulas for f. g, f + g, and f – g
(f + g) (x) = f (x) + g (x) = sin x + √4-cosx
(f – g) (x) = f (x) – g (x) = sin x – √4-cosx
f.g) (x) = f (x) . g (x) = sin x . √4-cosx
(f/g) (x) = f (x) / g (x) = sin x / √4-cosx
R’s domain is f, and that of g’s domain is also “R”. As – 1 cos x 1 for each x ε R, √4-cos x is defined for each x of the values of x ε R.) Hence, the domain of f + g, f.g, and f – g is R. Also, because 4-cosx > 0 for each x ε R, then the domain is f/g is R.
Occasionally we write f² to represent f.f. For example, f(x) = 8x, then
f² (x) = f.f (x) = f (x) . f(x) = 8x . 8x = 64x²
In a similar way, we give the denotation of f².f by f³. f by f & so on.