iterate' :: (a -> a) -> a -> [a]

I am trying going to go ahead and write my own iterate function.  But before I do I want to be clear on types.

Looking at  :: (a -> a) -> a -> [a]

The first part is (a -> a)  Now because it is in parentheses it is a function?

I can call iterate like this:

take 5 $ iterate (*2) 5

So (*2) is a possible function.  Does the brackets mean it is a function, the left hand a is indicating a general type and the right hand a means the return type must be the same as the function type.  Eg in the case of (*2) the 2 is an Int so the function returns and Int?  Is my understanding correct?

How could this be better explained?

The last bit is easier to understand.  a -> [a] meaning a singleton of general type and [a] means a list of same type.

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