**Contents (Click to skip to that section)**:

- What is a Function?
- Function Notation
- How To Know if It’s a Function
- Operations on Functions
- Types of Functions: A to Z

## 1. What is a Function?

A *function *is a type of equation that has exactly one output (y) for every input (x). If you put a “2” into the equation x^{2}, there’s only one output: 4. Some equations, like x = y^{2}, are not functions, because there are two possibilities for each x-value (one positive and one negative).

**Note to Excel users:** A “function” in Excel is a predefined formula. All Excel built-in functions are also functions in the traditional sense (i.e. you’ll get one output for every input).

## 2. Function Notation

**Function notation** tells you that the equation you’re working with meet the definition of a function.

The most common function notation you’ll see is f(x), which is read aloud as “f of x”.

The “f(x)” is used

in place of the “y”in a formula; They mean the exact same thing. For example, instead of the more familiar y = 2x, you’ll see f(x) = 2x. There’s no difference between the two formulas, other than the different notation.

**Any letter **can be used instead of f. For example:

- g(x)
- h(x)
- z(x)

## Examples

- y = 2x + 4; solve for y when x = 2.
- f(x) = 2x + 4; solve for f(x) when x = 2.

The two formulas above are telling you the same thing, they are solved in the same way (plug in your x-value and solve), and they give you the exact same solution:

- y = 2x + 4 = 2(2) + 4 = 4 + 4 = 8
- f(x) = 2x + 4 = 2(2) + 4 = 4 + 4 = 8

**You might also see questions written like this:**

f(x) = 2x + 4; solve for f(2)

This means the same thing as:

f(x) = 2x + 4; solve for f(x) when x = 2.

## Why Use Function Notation?

So, if y and f(x) mean the same thing, why use function notation at all? It might seem arbitrary to use f(x) or g(x) instead of y, but it can help you separate different parts of a formula and work with them more easily. Function notation gives you more information, and more flexibility.

For example, the chain rule uses the function notation F′(x), f′(x), g(x) and g′(x). It tells you that those four parts need to be dealt with separately:

Let’s take out all the different notations and replace them with the more familiar “y”:

It certainly

*looks*simpler, but good luck with trying to solve it!

## Types of Functions: Names and Arguments

The function name is the letter that represents the function:

- g(x): The function name is “g”
- h(x): The function name is “h”
- z(x): The function name is “z”

The argument is the letter in parentheses. In all three of the above examples, the letter is “x”. You might also see “t” or any other letter:

- g(x): The argument is “x”
- h(t): The argument is “t”
- z(s): The argument is “s”

## 3. How To Know if It’s a Function

The vertical line test is a simple way to figure out if you have a function.

You could also use to “many to one” rule:

**Is a function**: “*many to one*“. This is saying if you have multiple x-values that map to one y-value — say, (2,9), (3,9) and (6,9) — then that still qualifies as a function. Put more simply, it’s okay for a function to have multiple coordinate points in a straight line from left to right.**Not a function**: “*one to many*“. In other words, let’s say you had one x-value that maps to many y-values. For example, — in coordinate notation — (2,1) and (2,10). If the first number (the x-value) repeats, then you do not have a function. To put that another way, if you have multiple coordinate points in a straight line up and down, then that’s not a function.

As a practical example of one to many, one person can have several children. However, one child can only have one biological mother (an example of a one to one relationship). To put this into a more mathematical context: Let’s suppose a set*S*of ordered pairs (x, y) represents the statement “x is the mother of y”. The set S is a one to many relationship, because multiple ordered pairs can have the same value for x (i.e. one mother x can have multiple children, y).

## Issues with One to Many and Many to One

Although the above guidelines are found in many textbooks, they are** deceptively complicated** to use, because some graphs that have the “many to one” situation aren’t necessarily going to be functions; There may be other places (i.e. a couple of other coordinate points) that connect vertically, therefor disqualifying it as a function. These “rules” can also be** difficult to remember** (is it the first number that can repeat? Or the second?). Sometimes, it’s practically impossible to figure it out without some heavy algebra or the use of a computer. That’s because even if you have a few coordinates—or even an equation—you might be missing just a single point (perhaps with a very large x-value) that makes your graph not a function.

## Why is this important?

You can’t do much with an equation in calculus if it isn’t a function. You could cut up a complex function into smaller, function-like pieces (called piecewise functions), but in essence, **calculus only works properly with functions**. If you don’t perform a vertical line test before doing some calculus, then your solutions can be misleading or just plain wrong.

## 4. Operations on Functions

There are four **operations on functions**:

- Addition,
- Subtraction,
- Multiplication,
- Division.

For the purposes of the following examples, I’ll use functions f(x) and g(x).

You don’t *have *to use “f” and “g”. That notation is somewhat arbitrary. The functions could be represented by *any *letters; The choice depends largely on the preference of a particular author or professor. For example: j(t), s(t) or h(t). You might also see time(t) instead of “x”.

## Examples of Operations on Functions

## 1. Addition

With addition, you can add together two or more functions. The formula is:

**(f + g)(x) = f(x) + g(x)**

Suppose we wanted to add the following two functions:

f(x) = x^{2}

g(x) = 4x + 6

To get the solution, plug the functions into the formula:

(f + g)(x) = (x^{2}) + (4x + 6) = x^{2} + 4x + 6

Combine like terms when possible. To illustrate, assume you want to add the following two functions:

f(x) = 10x + 1

g(x) = 12x – 3

(f + g)(x) = (10x + 1) + (12x – 3) = 22x – 2

## 2. Subtraction

Two or more functions can also be subtracted. The formula is:

(f – g)(x) = f(x) – g(x)

I will use the same values for functions f(x) and g(x) as in my first example above.

(f – g)(x) = x^{2} – 4x + 6

## 3. Multiplication

To multiply two functions, use the following formula:

(f · g)(x) = f(x) · g(x)

Using the same values for f(x) and g(x) as above results in the following solution:

(f · g)(x) = (x^{2}) · (4x + 6) = 4x^{3} + 6x^{2}

## 4. Division

Two functions can also be divided. The formula for the division is:

(f/g)(x) = f(x)/g(x)

Plugging the values from the running example, you get:

(f/g)(x) = x^{2}/4x + 6

When working with operations on functions, simply look at what the operation is telling you to do – whether that is an addition, subtraction, multiplication or division. Once you identify the operation solve by plugging the values of the functions into the above formulas.

## 5. Types of Functions: A to Z

Obviously, this is a **very long list.** Either scroll down to find the type of function you want to learn more about, click a letter in the A-Z list below, or press Ctrl + F on your keyboard to search for specific types of functions. As an alternative, you can use the Google search box that’s embedded on the site (at the top right of the page). If you don’t see the function you need listed here, post a comment and I’ll add it!

## Types of Functions from A to Z (Click on the function name for more information about the specific function):

Click to skip to that letter: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

**Click on the function name for more information about the specific function):**

## Types of Functions: A to O

- Abelian Functions
- Absolute Value Function
- Absolutely Integrable Function (Summable Function)
- Arcsin function
- Additive Function
- Affine Function: Simple Definition, Examples
- Airy Function
- Arithmetic Functions
- Auxiliary Function

- Basis Functions: Simple Definition
- Barnes G-Function
- Blancmange Function (Tagaki Curve)
- Barnes G-Function
- Bessel Function
- Beta Function
- Bijective Function
- Binomial Function
- Bounded Function
- Boxcar Function

- C0, C1 C2 Functions
- Ceiling Function
- Center Function (Triangle)
- Chi Function (Legendre’s Chi Function)
- Choice Function & The Axiom of Choice
- Clausen Function (Integral)
- Closed Function
- Cofunction
- Complementary Function
- Complex Function
- Composite Function
- Constant Function
- Continuous Function
- Continuously-differentiable function
- Concave Function
- Convex Function: Definition, Example
- Cosecant Function
- Cost Function
- Cotangent Function
- Covercosine Function (Coversed Cosine)
- Cubic Function

- Debye Function
- Decreasing Function
- Dedekind Eta Function
- Digamma Function
- Dilogarithm Function
- Dirac’s Delta Function
- Dirichlet Function
- Dirichlet Eta Function (Alternating Zeta Function)
- Discontinuous Function
- Displacement Function.
- Divisor Function

- Elementary Functions
- Elliptic Function
- Empty Function
- Entire Function (Integral Function)
- Error Function
- Euler’s Totient Function / Phi Function
- Excosecant Function
- Explicit Function
- Exponential Function
- Exponential Integral Function
- Exsecant Functions

- Gamma Function, Multivariate Gamma Function
- Gauge Function
- Generalized Function
- Gaussian Function
- Gauss Hypergeometric Function
- Gudermannian Function (Gudermann)

- H Function (Fox’s H-Function)
- Hankel Function.
- Harmonic Function
- Haversine Function
- Heaviside Function
- Holomorphic Function (Analytic Function)
- Homogeneous Function
- Hurwitz Zeta Function
- Hyperbolic Functions

- Identity Function
- Incomplete Beta Function / Integral
- Increasing Function
- Indicator Function
- Injective Function
- Inner Function
- Integrand Function
- Interpolation Function
- Interval Function (Interval-Valued Functions)
- Inverse Cosecant Function
- Inverse Secant Function
- Inverse Cosine Function
- Inverse Functions
- Inverse Trigonometric Functions
- Integrable Function

- Lambert W function (Omega Function)
- Left-Continuous Function
- Legendre Function
- Linear Functions
- Lipschitz Function
- Locally Integrable Function
- Logarithm function
- Log Gamma Function

- Mapping Function
- Metric Function
- Möbius Function
- Modulo Function
- Meromorphic Function
- Meijer G-function:
- Monotonic Functions
- Multivalued Function
- Many to One
- Monomial Function

- Named Function.
- Non-Decreasing Function
- Non-Increasing Function
- Nowhere Continuous Function
- Nowhere Differentiable Function
- nth Root Function

## Types of Functions (O to Z)

## Types of Functions: P to Z

- Parabolic Function
- Periodic Function
- Piecewise Function
- Polar Function: Definition, Examples (Lemniscate, Limaçon…)
- Polygamma Function
- Polynomial Functions
- Power Functions
- Position Function
- Parametric Function
- Partial Function
- Partition Function P, Q: Simple Definition, Examples
- Pentation Function
- Popcorn Function (Thomae’s function): Definition, Formula and Graph
- Prime Counting Function
- Profit Function
- Point Function
- Polylogarithm Function
- Pulse Function

- Ramp Function
- Rational Function
- Real Valued Function
- Reciprocal Function
- Rectangular Function
- Right Continuous Function
- Riemann Zeta Function
- Revenue Function

- Sawtooth Function (Wave)
- Scalar Function
- Secant Function
- Septic Function (7th Degree Polynomial)
- Set Function
- Sextic Function
- Sigma Function
- Sign Function
- Sinc Function
- Sine Function
- Singularity Functions
- Sinusoidal Function
- Smooth Function
- Special Functions
- Square Function
- Square Integrable Function (Quadratically Integrable)
- Square Wave Function (Pulse Wave)
- Step Function
- Struve Function
- Superadditive Function & Subadditive Function
- Surjective Function

- Tangent Function
- Target Function
- Tetration Function
- Total Function
- Transcendental Functions
- Triangle Wave Function
- Trigonometric Function
- Trinomial Function

- Unary Function
- Uniformly Continuous Function
- Univalent Functions
- Unit Ramp Function
- Univariate Function
- Universal Function

- Weakly Increasing Function
- Weierstrass Function
- Weight Function
- Weierstrass Functions
- Whittaker Function

## Types of Functions: Divisor Function

In calculus, when an author uses the term “divisor function”, it usually refers to a function by which another function is divided. However, there are specific types of divisor functions used mainly in number theory, including the Dirichlet and summatory divisor functions.

## 1. Divisor Function as a Divisor

In general mathematics, a “divisor” is defined as “…another number by which another number is to be divided” (Oxford). For example:

**In 12 ÷ 4 = 3, 4 is the divisor. **

This can be extended to function division in calculus. For example:

**In f(g) ÷ f(h), f(h) is the divisor function. **

Dividing functions is something that crops up now and again in calculus, especially as it relates to defining functions.

You’ll also see this type of function in the quotient rule:

## Dirichlet and Summatory (Sigma) Divisor Function

In number theory, the **Dirichlet divisor function** is a count of how many positive divisors some number “n,” has, including n and 1.

For example:

- If n = 10, then d(10) = {1, 2, 5, 10} = 4.
- If n = 25, then d(25) = {1, 5, 25} = 3

The Dirichlet divisor function is sometimes denoted with (*d*(*n*)). However, this terminology may be confused with another function, which equals the *sum *of the positive divisors of n, including n and 1. This is sometimes referred to as the **sigma function **(not to be confused with the Weierstrass sigma function) or **summatory divisor function** to distinguish it from the Dirichlet. Note that the two divisors differ in that the Dirichlet version is a **count **of how many, while the summatory divisor function is a **sum **of all divisors.

While some authors specify that the function is Dirichlet or Summatory, others do not. Make sure you read the authors intent, rather than guessing the meaning.

## Restricted Divisor Function

The restricted divisor function is defined as the sum of the divisors of n, excluding n. It is usually denoted as s(n).

## Notation Notes

The divisor function can be denoted by d(n), ν(n), τ(n) or Ω(n).

## Types of Functions: Metric Function

A **metric space** is a *set *taken together with a *metric *on that set. The “metric” is actually a function; one which defines the distance between any two members of the set. Often the members of metric space are called *points*; so we can say the metric defines the **distance between any two points. **

More formally, it is a set X, together with a metric function *d*, which assigns a real number (we can call this *d(x,y)*) to every pair x, y. There are some restrictions on what type of function we can call the ‘distance function’, so we’ll go through them below.

## Properties of the Metric Function

A metric (our function *d* above) has to satisfy a few important properties, but they are all fairly simple and intuitive.

- If, and only if, x = y, then d(x,y) = 0.
- d(x, y) = d(y, x) (always)
- d(x, y) + d(y, z) ≥ g(x, z) (this is called the triangle inequality)

What these axioms tell us is that:

- The distance from a point to itself is always 1,
- A distance from one point to another is always the same as from the second point back, and
- The third side of a triangle is always less than the sum of its two sides (or equal, in the case that all points are on a straight line).

## Examples of Metric Spaces

The metric space you may be most familiar with is the real numbers; there, the distance function is defined as d(x,y) = |y-x|.

Positive real numbers can also be defined as a metric space, with a distance function

**d(x,y) = |log(y/x)|.**

The set of all points on the floor in your room, with the distance between them defined as the measured distance in millimeters, is also a metric space.

## Metric Space Circle Example

The more familiar way to define a circle’s metric space is through a Euclidean formula. Where it becomes more interesting is where you add different geometries, like Taxicab geometry, which requires you to get from a to b along a grid (much like how a taxicab might get from a to b in New York City).

## Types of Function: Named Function

In calculus, a “**named function**” refers to one of the following:

- A function that is fully defined
- A function that is familiar (i.e. it has a given name, like the Gamma Function).
- A function that is defined in mathematical software.

## 1. Fully Defined Types of Function

A named function sometimes means a function that is** completely and fully defined**, sometimes using logic. For example, Dijkstra & Scholten (2002) introduce the following named function in their book *Predicate Calculus and Program Semantics*:

“[

f . Y≡Y]

with f, [according to an earlier definition], for any Z given by

[f . Y≡ ¬B∧P] ∨ (B∧wp. S . Z)]”

I’ve abbreviated the full definition here, because—because of previous definitions—the notation would take up half a post. But the point is, the authors left no stone unturned in fully defining the function.

**Logic notation notes:**

If you’re unfamiliar with logic symbols used above, here’s what they mean:

- ¬ is the negation symbol, which is sometimes written, equivalently, as ~.
- ∧ is the “logical and” symbol.
- ∨ is the “logical or” symbol.

## 2. Familiar Functions

The term **named function** is sometimes used simply to mean **a function that is familiar and recognizable.** Getting an “unfamiliar” function (one that doesn’t adhere to some kind of well-recognized format) into a “familiar” one (see: Types of functions for some examples) has many benefits. These include known derivatives, known integrals, and the ability to use software to manipulate the functions.

As an example, the following G-function appears on page 224 of *Fractional Calculus and Its Applications: Proceedings of the International Conference held at the University of New Haven*:

Although this is indeed labeled as a “G-Function”, it isn’t a “known” one. All that’s needed is to rewrite the expression in the brackets so that the function becomes a “named function.” This is similar to the idea of forcing expressions to be explicit functions, so that they can be manipulated algebraically.

## Named Functions in Software

In programming, named functions are **defined by you,** and depend on the data you’ve input into the software.

For example, let’s say you’ve input a list of children and those children’s mothers.

You might have a named function **mother(x).**

**mother TINA** results in **JILL**.

As another example, the following named function (sum-of-squares) takes two numbers as arguments and outputs the sum of their squares (Wailing, 2019):

(define sum-of-squares

(lambda (x y)

(+ (square x) (square y)))).

## Types of Functions: The Square Function

**Contents **(Click to skip to that section):

## 1. Definition of Square Types of Functions

The square function **squares all inputs**. The formula is

f(x) = x^{2}.

The graph (sometimes called a *square map*) is a parabola. The parabola is concave up (i.e. it looks like a cup). If you place a negative sign in front of the “x^{2}” (not just the x-value), you’ll get an upside down parabola (i.e. one that is concave down).

The square function only has one intercept: at the origin (i.e. x = 0).

The square function is the inverse of the square root function.

## 2. Domain and Range

The domain of the square function is the set of all real numbers. The range is the set of all non-negative reals, because squaring a number will always give a positive result.

The square function can also be **defined in terms of its domain and range**. It takes every real number in the domain, squares that number, and assigns it to the result in the range. The function gets its name because numbers are squared. For example if x = 4, then 4^{2} = 16.

A few more examples of function values:

- f(0) = 0 * 0 = 0,
- f(2) = 2 * 2 = 0,
- f(-3) = -3 * -3 = 9,
- f(p) = p * p,
- f(height) = height * height.

## 3. Square Function Derivative

The square function derivative is 2x.

This uses the power rule to differentiate exponents. The steps for finding the derivative (shown in the above image) are:

- Copy the number of the exponent, and place it in front, so f(x) = x
^{2}becomes f(x) = 2x^{2} - Subtract 1 from the exponent in the new equation from Step 1: f(x) = 2x
^{2 – 1}= f(x) = 2x^{1}= 2x

## Types of Functions: Unary Function

A **unary function** has one input and one output. For example, the simple function *f(x)*is a unary function. This class of functions is the one most commonly studied in general math and calculus, so most of the types of functions you deal with in beginning calculus are unary.

The term *unary *is usually implied; when you hear reference to “a function,” it usually means a unary function. Other functions are referred to by their specific name to distinguish them from ordinary (unary) functions. For example, binary function or empty function.

**Unary real functions** take one argument and have a domain of real numbers. The ramp function is an example.

## Unary Function in Set Theory and CS

In **set theory**, you can think of a unary function *f* as one which, when applied to an argument x, results in a juxtaposition of the two, as in *f*(*x*) (Tarksy & Givant, 1987). A unary function can also be thought in more simple terms as just a function that **maps element of A to elements of A**. A related term is a *unary operation*, which is defined on set * A* as

*→*

**A***.*

**A**In **computer science**, unary functions act in the same way, except they are defined as **function objects called with a single argument**. In functional programming, these functions are also called *monadic functions*.

## Use in Lambda Calculus

In **lambda calculus**, a purely theoretical form of calculus, every value is a unary function. Lambda calculus is a simple way of applying functions to arguments.

## Types of Functions: References

Desmos Graphing Calculator.

Albert, John. Course Notes on Additivity. Retrieved from http://math.ou.edu/~jalbert/courses/additive_functions_2.pdf on June 14, 2019.Buchman, A. & Zimmerman, R. (1970). Eleventh Year Mathematics. Retrieved September 24, 2017 from: http://files.eric.ed.gov/fulltext/ED046731.pdf

Clarke, E. Lecture 5: Predicate Calculus. Retrieved April 7, 2020 from: http://www.cs.cmu.edu/~emc/15-398/lectures/lecture5.pdf

Tarsky, A. & Givant, S. (1987). A Formalization of Set Theory Without Variables, Volume 41. American Mathematical Association.

Edsger W. Dijkstra, Carel S. Scholten. (2012). Predicate Calculus and Program Semantics. Springer Science & Business Media.

Oxford Lexico, Retrieved November 30, 2019 from: https://www.lexico.com/en/definition/divisor

Ross, B. (Ed.). (2006). Fractional Calculus and Its Applications: Proceedings of the International Conference held at the University of New Haven. Springer.

Shapiro, Harold N. Introduction to the Theory of Numbers. Page 70. Retrieved from https://books.google.mn/books?id=4aX9WH8Kw_MC on June 3, 2019.

Shirali, S. First Steps in Number Theory: A Primer on Divisibility

Thompson, S. & Gardner, M. (1914). Calculus Made Easy, 2nd Edition. The Macmillan Company.

Young, C. (2018). Precalculus, 3rd edition. Wiley.

Wailing, F. (2019) Session 5: Racket Functions. Retrieved December 3, 2019 from: https://www.cs.uni.edu/~wallingf/teaching/cs3540/sessions/session05.html

Calculus of a Single Variable

Wenpang, Z. (2009). Proceedings of the Fifth International Conference on Number Theory and Smarandache Notions (Shangluo University, China). Infinite Study.

Zhang, W. (2005). Research on Smarandache Problems in Number Theory (collected …, Volume 2.

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