New Functions from old

There are three primary ways in which new functions can be ``constructed'' from old ones. The first and most familiar one is integration. The second method is the inversion of a monotone function. The third is the implicit definition by an equation in two variables.

Formally, the integral of a function f is a function denoted by $\int$f which satisfies d ($\int$f )/dx = f; note that such a function is only determined upto a constant since a constant has derivative 0.

Similarly, if f (x) is a function, its formal inverse is a function g(y) so that gof = identity. Finally, if f (x, y) is a function of two variables, we can look for a function g(x) so that f (x, g(x)) = 0 identically.

Algebraically this is all we need. We can easily compute the values of derivatives at various orders of the new functions in terms of those of the old functions.

Analytically, we need to show that such functions exists under certain reasonable conditions on the given data. For exmaple, we have already constructed an integral for a polynomial function.

Here is an example of inversion:

Exercise 44   Let f (x) = x³ show that there is a function g(y) so that g(f (x)) = x for all x.

And an example of an implicit function:

Exercise 45   Let f (x, y) = x² + y² - 1 show that there is a function g(x) for 0 < x < 1/2 so that f (x, g(x)) = 0.

In the sections below, we shall construct such solutions in greater generality.