Concept of function

If you are given one angle a, you can immediately compute its complementary angle b in a right-angled triangle: b = 180 - a. You can then set up a table

and enlarge it as required. It allots to every value of a a value of b. This is the general feature of a function, namely that of a mutual relationship; it is represented most simply by a table. a and b are the variables. If you give a a value, you can compute b; a is then called the independent and b the dependent variable. You can invert this relationship and make a into the dependent, b into the independent variable. Quantities which do not vary are constants.

As a rule, the independent variable is denoted by x, the dependent variable by y, so that b = 180 - a becomes y = 180 - x.

y is a function of the variable x whenever
definite values of y are ascribed to the values of x by a certain rule .

While a table can only represent a function with gaps, its equation comprises all its values. Functions differ greatly from each other: y=2x+4, y=6x²+2x+8, etc. All of them should have the form y = f(x) (say: y equals f of x). If you shift all terms to one side: y - 6x² - 2x - 8 = 0, you obtain an implicit function F(x,y) = 0. Obviously, the implicit function F(x,y) = 0 yields an explicit function y = f(x). The theory of functions is called analysis.

You encounter functional relations in all activities. The length l(y) of an iron rod is determined by the prevailing temperature t(x); there corresponds to every temperature t over a certain range a definite length l, and to every value of t, a value of l. Hence the length l is a function of t: l = f(t); the interest z of a certain amount of money loaned for a given time is a function of the interest rate p: z = f(p), etc.

Functional equations take their place beside determining equations and identities. The difference between the determining equation mx + n = 0 and the functional equation y = mx + n is that x in the determining equation represents a definite value, in other equations even several values, while in the functional equation x is variable and definite values are allotted to it. Then there is the additional type of equation, the identity, for example: y + x = x + y, in which all entering numbers can assume arbitrary values. Every arithmetic rule can be represented by an identity. We can now use these three types of equations to focus on the concepts of arithmetic, algebra and analysis:

Arithmetic is the theory of identities,
Algebra that of determining equations,
Analysis that of functional equations.

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