2f

2f denotes the function obtained by multiplying the values of a given function f by 2. If f is a function from a domain X to a codomain Y, then 2f is the function 2f: X → Y defined by (2f)(x) = 2 f(x) for all x in X. This is a standard example of scalar multiplication of a function, leaving the domain unchanged while scaling its outputs by the factor 2.

Properties of 2f follow from those of scalar multiplication. For any functions f and g with the same domain and any scalar a, (af)(x) = a f(x) for all x, and (f + g)(x) = f(x) + g(x). Consequently, 2f + 2g = 2(f + g). The derivative and integral behave similarly: (2f)' = 2 f', and ∫(2f)(x) dx = 2 ∫ f(x) dx. For composition, (2f) ∘ g = 2 (f ∘ g) since (2f)(g(x)) = 2 f(g(x)).

Graphically, multiplying a real-valued function by 2 vertically stretches its graph by a factor of 2. If f takes vector values, 2f scales each component by 2.

Notational note: 2f is not the same as f², which would denote the pointwise square (f(x))². In other contexts, 2f can be seen as shorthand for other concepts (for example, two-factor authentication is commonly abbreviated 2FA), but this article focuses on the mathematical usage of 2f as a scalar multiple of a function.