2.2.4 Functions
Notation to indicate that a function f maps elements of a set A to elements of a set B is:
[2.4]

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
[2.5]

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:
[2.6]

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.
[2.7]

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