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 f |a. The former is read as “f of a“. The latter is read as “f evaluated at a“.
Consider f : p → m and g : n → p. The composition of f and g is the function f ◦ g 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 : n → m is the matrix of its first partial derivatives.
Note that the Hessian of a function f : n → is the Jacobian of its gradient.