Map

Description:

• Refer to vector-valued functions
• Maps are linear functions that return a vector of value instead of 1 value
  • $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$
• The components of the map $f$ are scalar-valued functions $f_i$ for $i=1,2,...,m$

Sets related to functions:

• Graph:
  • Consider function $f:{\mathbb{R}}^n:\mathbb{R}$
  • The graph of $f$ is the set of input-output pairs that $f$ can attain
    • with $f(x)$ as the $n+1$ element of the vector
  • graph $f=\{(x,f(x))\in {\mathbb{R}}^{n+1}:x\in {\mathbb{R}}^n\}$
• Level set, or contour line:
  • The set of points that achieve exactly some value for the function $f$
  • For $t\in \mathbb{R}$, the $t$-level set of the function $f$ is defined as:
    • $C_f(t)=\{x\in {\mathbb{R}}^n:f(x)=t\}$ but typically only take 2 $x(i)$ to graph
    • 
  • The $t$-sublevel set of $f$ is the set of points that achieve at most a certain value for $f$:
    • $L_f(t):\{x\in {\mathbb{R}}^n:f(x)\le t\}$