Functions of vectors and vectors of functions

Question

In page 22 of the Matrix Calculus For Deep Learning, the authors wrote: It is the nature of neural networks that the associated mathematics deals with functions of vectors not vectors of functions. Did it mean f(x1, x2, ..., xn) as opposed to (f1(x), f2(x)..., fn(x))? Can you give a relevant example of (f1(x), f2(x)..., fn(x))? Thanks.

Answer 1

In neural networks we generally have multiple inputs entering a perceptron and one output. Therefore, the output is a function of the inputs, e.g. $f(x_1,..,x_n)=\sum_{i=1}^{n}w_ix_i$ where the $w_i$ are the weights associated to each input. Here the multiple inputs can be seen as a single vector of inputs, thus, $f$ is a function of a vector.

A vector of functions occurs when one input parametrizes a set of outputs. We see this in parametric curves. For example $f(x)=(\cos{x}, \sin{x})$ parametrizes a circle.