1. Vector Valued Functions
   Any function which maps into Rⁿ, where n does not equal 1 is often referred to as a Vector Valued Function because the 'output' can be perceived as as vector in the appropriate Rⁿ.
   Clickable Examples
   1. An example from R to R²
   2. An example from R² to R³
   3. Another example from R² to R³
2. Graphs
   The word "graph" that you are so familiar with has typically meant the set of ordered pairs in R² of the form (x,f(x)). In fact, this is a special case of something more general. The word 'graph' is any ordered n-tuple of the form (domain element, f(domain element)). For maps from R to R, this would consist of an ordered pair. But if the domain were R² and hence of the form (x,y), and the range were in R, your graph would be of the form (x,y,z) where z=f(x,y) and hence reside in R³.
   Examples of Graphs
   1. The graph of a map from R to R² resides in R³. (Click here to see 3D graphs)
   2. The graph of a map from R² to R² resides in R⁴. It therefore can't be visualized.
   3. The graph of a map from R^m to Rⁿ resides in R^{n + m}. If n + m > 3, it also can't be visualized.
3. Parametric Curves and Images
   A parametric curve is one where the 'inputs' or 'parameters', though used to determine the outputs, are living in another place, or space that we are not interested in visualizing. The easiest example to speak of would be one where the parameter in question is time (t). You might have functions x=x(t), and y=y(t) (or z=z(t)) that vary with time, and your interest would be to see how (x,y)=(x(t), y(t)) move around in the plane (or how (x,y,z)=(x(t), y(t), z(t)) flies around in space.) In case a concrete example from your past would help, consider the parametric definition of a circle given by (cosA,sinA), where A is the angle formed between a radius in the unit circle and positive x axis.
   INSERT HERE A FEW EXAMPLES OF PARAMETRIC CURVES
   Often we can think of a parametric curve in Rⁿ which depends on m parameters as a map from R^m to Rⁿ. We do not include a visualization of the parameter(s) but rather just the set of outcomes in Rⁿ. This picture of the outcomes is therefore not a graph. We call this resulting visualization the "image" of the map, or function. One can look at the image of a function as well as the image of a parametric curve but in the latter case, we generally only consider images. To make this clearer: ou are used to looking at the graph of a position function vs. time. These two notions are related because one typically only plots the "image" of a paramteric curve. The image resides in the co-domain. We do not often look at images of functions when we can picture their graphs (ie: when the graph can fit in 3 dimensions or fewer), yet we can choose to only look at images particularly when the parameter is time.THIS EXPLANATION BLOWS AND I MUST FIX IT.
   1. An example from R to R²
   2. An example from R to R³

4. Exercises

   1. Level 1
   2. Level 2
   3. Level 3

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