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


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


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

Continuous Functions

  • 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
  • function-multi.png

Function Spaces

Functions, like Vectors, can form Vector Spaces