1.5 Other Functions Previous Section Next Section

Are there other functions?
Yes but we will mostly be concerned with standard functions.

What are the other functions we may meet?

Piecewise standard functions: functions that are standard in subintervals of the domain but not necessarily the same ones in all.
Functions defined by infinite series in particular by a series of powers xn with coefficients that are standard functions of the powers.
Functions defined using the operations of calculus.
Functions defined recursively or implicitly:
This means a function that is described by a procedure for constructing its values that requires repeated application in order to define them.
For example, the Fibonacci numbers, f(n) form a sequence according to the following rules: f(0) = f(1) = 1; f(n) = f(n - 1) + f(n - 2) for integers n greater than 1.
This is a recursive definition.
Functions that arise from real phenomena.
These usually start off being unknown, They may be anything. It is remarkable how well we can do in treating them as if they were standard functions, or were in one of the other classes of functions described above.

A sequence, can be considered a  function defined with N or a subset of N as domain.
We can add subtract or multiply sequences, or functions defined on the same domains, and divide one by another as long as the one divided by is not 0.

Why consider standard functions?
They are available on calculators and computers.
We can describe most applications using them.
They have only isolated singularities.
They are infinitely differentiable over most of their domains.except at singularities.