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"The" exponential function is ex. A more general exponential function is any function of the form AeBx, for any non-xero constants "A" and "B". Alternately, Any function of the form CDx (for constants "C" and "D") would also be considered an exponential function. You can change from one form to the other.
You can represent any given function in as many different ways as you want.
An arithmetic function is any function which is defined for all positive integers, and has values which are either real or complex.
No. If the range of the first function is not the domain of the second function then the composite function is not defined.