Here's a pretty trivial example in Haskell - computing the factorial function using a mutable variable.

    import Control.Monad.ST
    import Data.STRef

    fact :: Int -> Int
    fact n = runST (fact' n)

    fact' :: Int -> ST s Int
    fact' n = do a <- newSTRef 1
                 mapM_ (\x -> modifySTRef a (*x)) [1..n]
                 readSTRef a

Here the function `fact'` uses mutable variables (encoded in the use of `ST` in its type -- `ST` stands for State Thread) but the function `fact` is pure -- the call to `runST` ensures that none of the side effects leak out of `fact'`.

As with most Haskell code, the types are optional - I included them for clarity.