Operations on Functions
Scalar Multiple of a Function
Let the function f : x → 3x2 + 1, for all of x ∈ R. The function g: x → 2 (3x2 + 1) for all of x ∈ R is like that g (x) = 2f (x), for all of x ∈ R. We declare that g = 2f and that g is a scalar multiple of f through 2. In the above instance, there is nothing special regarding the number 2. We could have taken up any real number to create a new function from f. Also, there is nothing special regarding the particular function which we have considered. We could as well as have taken up any other function. This recommends the following definition: Assume f be a function with domain D & let k any real number.
The scalar multiple of f by k is a function along domain D. It is indicated by kf and is described by setting (kf) (x) = kf (x).
Two special cases of the above definition are significant.
(a) Given particular function f, if k = 0, the function kf turns out to be the zero function. That means, kf = 0.
(b) If k = - 1, the function kf is say the negative of f and is indicated simply by - f.