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basicfunctions_1013 - THE BASIC FUNCTIONS WE WILL ENCOUNTER...

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THE “BASIC” FUNCTIONS WE WILL ENCOUNTER IN MATH 1013 REVIEW/CONTEXT As discussed in class, every function, no matter how complicated, can be built up from a small number of basic function steps (steps which cannot be reduced any further in terms of even simpler sub-steps, or which we understand so well we don’t bother to break them down further in this way), combined with the operations of addition, subtraction, multiplication and/or division (“arithmetic” steps); composition (taking the output of a previous step and using it as input to the next step); inversion (forming the inverse function of an existing function, if the existing function has an inverse). In this section of the notes we will list, and discuss briefly, all of the basic function steps that you will meet in the course. Before we do so, it is worthwhile expanding a bit on the last entry in the “complicating steps” list above. Students who want to get to the main topic right away, can skip over the aside which follows, and come back to it later. ASIDE REGARDING THE INTERPRETATION OF THE INVERSES The third step in the “complicating steps” list above is not stated explicitly in most textbooks. This is probably because the inverses of basic functions are usually in- terpreted as separate basic functions themselves in such textbooks. However, more complicated functions can also have inverses, so it is good to include this step explic- itly in our list. Remember that the inverse of a function y = f ( x ) (if such an inverse exists) means a function, denoted f - 1 , such that, if y = f ( x ), then x = f - 1 ( y ); f - 1 ( y ) does NOT mean 1 /f ( y ).
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As discussed previously, the inverse of a function is generally a different function than the original function, i.e., for f - 1 , the rule for getting from the “new” input (independent variable) value, y , to the “new” output (dependent variable) value, x , is different than the rule, relevant to f , for getting from the “original” input (independent variable) value, x , to the “original” output (dependent variable), y . For example, the cubing function, y = x 3 is such that x 3 1 6 = x 3 2 whenever x 1 6 = x 2 , and hence y = f ( x ) = x 3 DOES have an inverse. This inverse is called the cube root function, which we write in this context as x = f - 1 ( y ) = y 1 / 3 . To a mathematician, f and f - 1 , though different functions, contain exactly the same information (the linking of certain pairs of x and y values, the association being from x y for f and from y x for f - 1 ). As soon as one knows f , one also knows f - 1 (provided the inverse actually exists). Thus one can think of f - 1 as a “new” function constructed from the “old” function, f . A mathematician thus typically thinks of f - 1 as a non-basic function since it is constructed from a simpler original function f by the step of “inversion” and hence will be inclined NOT to include, e.g., the cube root function in the list of “basic” functions.
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