A function is a rule that associates inputs with outputs.

They form the core of many aspects of mathematics and numerous programming languages.

The fundamental explanation of functions comes from Set Theory, in which a function is regarded as the mapping from one set, AA, to another set, BB, expressed as:

f:ABf : A \rightarrow B

Functions are commonly denoted using the letters ff, gg, ii, or jj.

Consider a function, ff, that maps a set of people's names to their ages:

  • A={Sarah,Geoff,Clyde,Betty}A = \{\text{Sarah}, \text{Geoff}, \text{Clyde}, \text{Betty}\}
  • B={0,1,2,,120}B = \{0, 1, 2, \cdots, 120 \}
  • f:ABf : A \rightarrow B

An example of the above mapping

The set of possible inputs is called the Domain of a Function or DfD_f:

Df=A={Clyde,Sarah,Geoff,Betty}\color{lightblue}D_f = A = \{Clyde, Sarah, Geoff, Betty\}

The set of possible outputs is the co-domain or coDfco-D_f of the function.

coDf=B={0,1...120}\color{orange}coD_f = B = \{0, 1 ... 120\}

In this case, the co-domain includes all plausible human ages.

Given the input set AA, the set of possible outputs is {11,34,98}\{11, 34, 98\}. This set is known as the range of the function ( RR ).

R={11,34,98}\color{darkred}R = \{11, 34, 98\}

To show a single input-output relationship, we could write:

f(Sarah)=34f(Sarah) = 34

The output 34 is the function's image, and the corresponding input, SarahSarah, is the pre-image.

Functions are considered a "well-behaved relation". That means that for each input, there must be exactly one output. This example qualifies as a function because each person has a unique, valid age.

We can express functions as the relationship between an input variable and its output. For example, the function to convert temperature in Fahrenheit to Celsius is as follows:

f(x)=(x32)×59{f(\text{x}) = (x - 32) \times \frac{5}{9}}

The complete definition of a function should include its domain and co-domain. Since Fahrenheit and Celsius are real numbers, so we would define the function using the Special Infinite Set R\mathbb{R}.

f:RRf : \mathbb{R} \rightarrow \mathbb{R}

The two parts combined give the complete definition of the function:

Let f:RRf: \mathbb{R} \to \mathbb{R}, f(x)=(x32)×59f(x) = (x - 32) \times \frac{5}{9}

In programming languages, mixing the type declaration with the implementation is common. Below is an example of the function ff in Python. It takes an input xx as a float and returns a float, described using the notation -> float.

def f(x: float) -> float:
    return (x -  32) * (5/9)

Plotting Functions

We can create a set of input values and their corresponding outputs, then visualise them geometrically by drawing the inputs and outputs on the x-axis and y-axis, respectively. This visualisation is called a graph of a function.

Here is a plot of the Fahrenheit to Celsius function earlier, plotted across a range of inputs: from -100 to 100.

Fahrenheit to celsisus function plot

When the graph is a straight line like this, it's called a Linear Function. There are other names for common function types:

  • Linear Function: A function where the output is proportional to the input.
  • Quadratic Function: A function where the output is proportional to the square of the input.
  • Exponential Function: A function where the output is proportional to a fixed base raised to the power of the input.
  • Polynomial Function: A function that we represent as a sum of terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. Linear and quadratic functions are specific types of polynomial functions.

There are some other important properties of functions:

One-to-one / Injective

We consider a function "one-to-one" or "injective" if each output is associated with exactly one input and no two different inputs have the same image.

Onto / Surjective

A function is "onto" or "surjective" if every element in the co-domain is output for at least one input in the domain.


We call a function Bijective if it is both injective and surjective.


A function is continuous at a point x = c under the following conditions:

  • f(c) is defined.
  • The limit of f(x) as x approaches c exists.
  • The limit of f(x) as x approaches c is equal to f(c).

That is, limxcf(x)=f(c)\lim_{x \rightarrow c} f(x) = f(c)

A function is discontinuous at a point x = c if any of the above conditions are not met.

A function might only have discontinuatities specific internals

Some special cases apply:

  • Polynomials are always continuous.
  • Rational functions: Continuous when the denominator is not zeo.
  • Trig functions: continuous on their domain.
  • Exponential and log functions: continuous when defined.