N-fold smoothed function

The idea of smoothing a function is an important concept in signal processing. If f is a function and dx is some small number, then the smoothed version of f is the function whose value at a point x is the average of f(x - dx), f(x), and f(x + dx). Write a procedure smooth that takes as input a procedure that computes f and returns a procedure that computes the smoothed f. It is sometimes valuable to repeatedly smooth a function (that is, smooth the smoothed function, and so on) to obtained the n-fold smoothed function. Show how to generate the n-fold smoothed function of any given function using smooth and repeated from exercise 1.43 .

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(define (square x)
  (* x x))

(define (cube x)
  (* x x x))

(check-equal? (round ((smooth square) 3)) 9.0)
(check-equal? (round ((smooth cube) 10)) 1000.0)