# Recap: A more formal understanding of functions

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This is a short summary of the key terms from the excellent video, “A more formal understanding of functions”, from the Linear algebra section on Khan Academy.

At its core, a *function* is just a mapping of one set to another set.
So, for example, for every element in set X, we associate an element
in set Y. And then we call this association a *function*.

The origin set is called the *domain*. The destination set is called
the *co-domain*. These sets are part of the function definition,
although oftentimes people don’t mention them.

A more concrete example might be

\[f(x) = x^2\]Alternatively, this may be written as

\[f: x \mapsto x^2\]The second notation perhaps expresses the association better. This
function maps the *domain* of real numbers to the *co-domain* of real
numbers. Or, in other words, for this *function*, we can put in any
real number and it will give us a real number back.

The subset of the co-domain that is actually mapped is called the
*range*. So, for example, if we had a function g, which takes in any
two-dimensional real number and returns 2, the codomain may be the set
of real numbers, but the range would be limited to 2.

## Vector Transformations

When the co-domain is a one-dimensional real number, the function is
called a *scalar valued function* or a *real valued* function. The
example above is a *scalar valued function* because $f: x \mapsto x^2$
maps $\mathbb{R} \mapsto \mathbb{R}$.

When the co-domain maps to multi-dimensional real number, the function
is referred to as a *vector valued function*. Or, in other words, when
the co-domain is $\mathbb{R}^m$, where $m > 1$, this is a *vector
valued function*. People sometimes refer to this as a vector
*transformation*, but essentially a *transformation* is simply a
*function*.

## Key Terms

- function
- domain
- co-domain
- range
- scalar valued function
- vector valued function
- transformation

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