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A linear function when graphed takes the form of a straight line.
"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.
If you are asking about periodic function in the real world, than there are millions of examples. The first one that comes to mind are the sin and cos functions. These are used in almost every field of science.
Well, if you solve the equation for "y", you have "y" as a function of "x". Or you can do it the other way round; solve for "x", to get "x" as a function of "y" (the first option is more commonly used, though).