On The Definition of Functions
Functions or rules for calculating them can be introduced or defined in many
ways. Saying how to compute a number or quantity from other numbers and
quantities, or geometric objects, gives a computation rule and hence a
function.
- In algebra, you may have seen the definition of functions (computation
rules) using polynomials, square roots, small powers and so on.
- In trigonometry, you may see the right triangle and unit circle
definitions of the sine, cosine and tangent functions. Again, in
trigonometry, you may see the idea of the inverse to a function employed, to
define or introduce the arccos, arcsin and arctan functions.
- The usual set theoretic approach to defining a real-valued function f(x)
of a real variable x is to give a set of ordered pairs, a graph with the
vertical line property. Here a finite set of ordered pairs (with the
vertical line property) correspond to a table of values for a function or a
finite set of points in the plane.
- The definition of real-valued functions of a real variable using finitely
many arrows or a table with finitely many entries can be easily be converted
into sets of ordered pairs with the vertical line property.
- An alternate set theoretic approach to defining a real-valued function f(x)
of a real variable x is to give a set of ordered pairs with the
horizontal line property. Here a finite set of ordered pairs (with the
horizontal line property) correspond to a table of values for a function.
The usual or standard approach follows by transposing the set or more
precisely the coordinates of all its points to obtain the transposed set.
The latter then gives the graph of the function.
- In differential calculus, a function g(x) was given,
introduced or defined, by the slope or derivative f¢(x)
of another function f(x).
- In integral calculus, a function F(x) may be
introduced or defined as the area-under-a curve between two points.
The natural logarithm ln(x) in particular can be defined as the area under
the curve s = 1/t from t =1 to t = x.
The foregoing extends the last
section in Chapter 18, Volume 3, Why
Slopes and More Math , a book on calculus etc.
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