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