A function is something that maps an input (domain element) to a unique output (range element).

  • If the output is not unique for a given input element, then we have a Relation

Domain and Range

  • The domain of a function is the set of allowable values for the independent variable.
  • The range is the set of possible values for the dependent variable.

Composition of Functions (Function Composition)

There is also this idea of Relational Composition

Inverse Functions

We say that a function is an inverse of a function if for any in the domain of .

inverses are unique.

Note that the inverse of f is NOT the same thing as the reciprocal of f.

Invertibility: Not every function possesses an inverse. For the inverse to be a true function, it must have a single output for every input. The requires that the original function is one-to-one, i.e. an Injective Function

in STAT206, to get an inverse, swap the two variables and solve. you want to write in terms of You know that , so just swap the variables, you have , so you get

In general, you let , so if (this is invertible), you have (so simply swap the variables from the original function), and solve for , you get . The final answer is

Classifications of Functions


  • Algebraic function: produced by taking sums, products, and quotients of roots of polynomials and rational functions
  • Any function that is not algebraic is called transcendental. Exponential and trigonometric functions are examples.

Categorization of Functions

From SE212, TODO add notes with Binary Relation