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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.
A relation is a function when an x value only has one y value associated with it. An easy way to tell this is to graph the relation, then draw a vertical line through it. If, at any point, it touches the graph twice, the relation isn't a function.
A function cannot be a probability mass function (PMF) if it violates the properties of a PMF. A PMF must assign a non-negative probability to each possible outcome of a discrete random variable, and the sum of probabilities for all possible outcomes must be equal to 1. If a function does not satisfy these properties, it cannot be considered a PMF.