Function
In mathematics, a function $f(\cdot)$ is a set of pairs $x, f(x)$, where
- $x$ is input, $f(x)$ is output
- all possible inputs $x$ that a function can take is the domain of $f$
- all possible outputs $f(x)$ of a function $f(\cdot)$ is the range of $f$
Operations on Functions
Composition
For two functions $f$ and $g$, composition is $f \circ g$
- $(f \circ g) (x) = f(g(x))$
- $g$ is applied first, then $f$
For example,
- $\sqrt{1 - x^2}$
- $g = x^2$, it's inside
- $f(t) = \sqrt{1 - t}$, it's outsize
Inverse
For $f$ its inverse is $f^{-1}$
- $x = f^{-1}(x)$ if $f(f^{-1}(x)) = x$
- $f^{-1}(\cdot)$ is a function that "undoes" $f(\cdot)$
Single Variable Functions
That's the simplest type of functions: they have one input and one output
Important Function Classes
- Functions are continuous is their Limits always exist
- otherwise functions are discontinuous
- these are important functions in Calculus
Multi Variable Functions
These functions are more complex:
- they can have multiple inputs and multiple outputs
-
Functions, like Vectors, can form Vector Spaces
Sources