Functions and maps

Functions

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

    \[f: \mathbb{R}^n \rightarrow \mathbb{R},\]

to refer to a function with “input” space \mathbb{R}^n. The “output” space for functions is \mathbb{R}.

Example: The function f: \mathbb{R}^2 \rightarrow \mathbb{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: \mathbb{R} \rightarrow \mathbb{R}, with values f(x)=\log x if x>0, and -\infty otherwise. The domain of the function is thus \mathbb{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

    \[f: \mathbb{R}^n \rightarrow \mathbb{R}^m,\]

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

Example: A map.
The map f: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2} with values

    \[f(x) = \begin{pmatrix} \sqrt{x_{1}^2 + x_{2}^2} \cos(x_{3}) \\[0.7em] \sqrt{x_{1}^2 + x_{2}^2} \sin(x_{3}) \end{pmatrix}.\]
has components the functions f_{i}: \mathbb{R}^{2} \rightarrow \mathbb{R}, \; i=1,2, with values

    \[f_{1}(x)=\sqrt{x_{1}^2+x_{2}^2}\cos(x_{3}), \quad f_{2}(x)=\sqrt{x_{1}^2+x_{2}^2}\sin(x_{3}).\]

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