2.2.4 Functions

2.2.4  Functions

Notation to indicate that a function f maps elements of a set A to elements of a set B is:


A is the function’s domain; B contains its range. We are primarily interested in three types of functions:

  • functions from to ,
  • functions from n to ,
  • functions from n to m.

We call functions of the first form real—they map real numbers to real numbers. The natural logarithm function is a real function, which we denote log. We do not employ the logarithm base 10. If a function f has an inverse, we denote this f –1. The derivative of a real function f may be indicated with differential notation or simply as f ′.2 We indicate the value of a function f at a particular point a as either f(a) or |a. The former is read as “f of a“. The latter is read as “f evaluated at a“.

Consider : pm and : np. The composition of f and g is the function fg from n to m defined as


The gradient f and Hessian 2f of a function f : n are the vector of its first partial derivatives and matrix of its second partial derivatives:


The Hessian is symmetric if the second partials are continuous.

The Jacobian of a function f : nm is the matrix of its first partial derivatives.


Note that the Hessian of a function f : n is the Jacobian of its gradient.