Functions and maps

L. El Ghaoui

Functions and maps

Functions

In this course we define functions as objects which take an argument in $\mathbf{R}^n$ , and return a value in $\mathbf{R}$ . We use the notation

$f: \mathbf{R}^n \rightarrow \mathbf{R},$

to refer to a function with “input” space $\mathbf{R}^n$ . The “output” space for functions is $\mathbf{R}$ .
Example: The function $f: \mathbf{R}^2 \rightarrow \mathbf{R}$ with values

$f(x)=\sqrt{\left(x_1-y_1\right)^2+\left(x_2-y_2\right)^2}$

gives the distance from the point $\left(x_1, x_2\right)$ to $\left(y_1, y_2\right)$ .
We allow for functions to take infinity values. The domain of a function $f$ , denoted $\operatorname{dom} f$ , is defined as the set of points where the function is finite.
Example:
– Define the logarithm function as the function $f: \mathbf{R} \rightarrow \mathbf{R}$ , with values $f(x)=\log x$ if $x>0$ , and $-\infty$ otherwise. The domain of the function is thus $\mathbf{R}_{++}$ (the set of positive reals).

Maps

We reserve the term map to refer to functions which return more than a single value, and use the notation

$f: \mathbf{R}^n \rightarrow \mathbf{R}^m,$

to refer to a map with input space $\mathbf{R}^n$ and output space $\mathbf{R}^m$ . The components of the map $f$ are the (scalar-valued) functions $f_i, i=1, \ldots, m$ .
Example: a map.

Functions and maps

Functions

Maps

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