1.5 Other Functions

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: these are functions that are standard in subintervals of their domains but not necessarily the same standard function in all of them. The function that is 0 for any negative argument and 1 for positive arguments is an example. This is called a step function.
The absolute value of x, which is -x for negative argument x and +x for positive x is another. Its graph looks like a V centered at the origin in x.

Functions defined by infinite series: in particular by a series of powers xⁿ with coefficients that are standard functions of the powers. A simple and fundamental example is the geometric series, defined by g(x) = 1 + x + x² + ... + x^k + ...

Functions defined using the operations of calculus: these are typically functions defined as derivatives or integrals of standard functions. Such definitions can easily be made once those concepts are defined.

A sequence can be considered a function defined with N or a subset of N as its domain.

Functions defined recursively or implicitly: a recursive definition of a function is one that describes it by a procedure for constructing its values that requires repeated application in order to define them over its entire domain.

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 of this sequence.

Implicitly defined functions will be discussed in detail in section 1.7.

Functions that arise from real phenomena: these usually start off being unknown. They may be anything. It is remarkable how well we can do by treating them as if they were standard functions, or as if they were in one of the other classes of functions described above.

Why consider standard functions?

They are available on calculators and computers.
They have only isolated singularities.
They are infinitely differentiable over most of their domains, except at certain singular points which are usually easy to locate.
They can be defined in the complex plane.
They are immensely useful.

We can add, subtract or multiply any sequences, or functions that are defined on the same domain (whether they are standard or not), and divide one by another wherever the one divided by is not 0. To do so, at each argument in their domain, add, subtract, multiply or divide their values.