How do I know a function or an operation in R is vectorized?

Vectorization in R basically means that any looping is moved to a faster, compiled language such as C or FORTRAN. For that to occur the vector(s) in question should be "atomic" - i.e. it should be "flat" and homogeneous - and the vector type, which you can check with typeof(), should make sense for the operation(s) being performed. If it is atomic then it is vectorized.

You can check if a vector is atomic using is.atomic(). Another type of vector that is not vectorized is called "recursive", which you can check using is.recursive(). Recursive objects can contain other objects of any type, i.e. they can be heterogeneous. Lists and data frames are recursive.

Try something like the following to gain some insight into atomic vs. recursive:

# Atomic:
1
1:3
c("a", "b", "c")
c(T, F, T)

# Recursive:
list(nums = 1:3, letts = c("a", "b", "c"), logics = c(T, F, T))
data.frame(nums = 1:3, letts = c("a", "b", "c"), logics = c(T, F, T))

# Vectors can be atomic or recursive:
is.vector(1:9) # TRUE
is.atomic(1:9) # TRUE
is.recursive(1:9) # FALSE

is.vector(list(nums = 1:9, chars = "x")) # TRUE
is.atomic(list(1:9)) # FALSE
is.recursive(list(1:9)) # TRUE

# Matrices are atomic, data frames are recursive:
is.vector(matrix(1:9, 3)) # FALSE
is.atomic(matrix(1:9, 3)) # TRUE
is.recursive(matrix(1:9, 3)) # FALSE

is.vector(as.data.frame(matrix(1:9, 3))) # FALSE
is.atomic(as.data.frame(matrix(1:9, 3))) # FALSE
is.recursive(as.data.frame(matrix(1:9, 3))) # TRUE

I think you can assume that many, if not most, of the R functions that you use most frequently are vectorized. I don't think there is any way to check this other than by looking at the documentation or the function internals. Whenever you think about writing a for loop to do simple element-wise operations, think about how to do it using vectorization. With enough practice it will become second nature to you. For more details I can recommend this blog post from Noam Ross.