# Mathematical Function Family

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A Mathematical Function Family is a function family of mathematical function.

**Example(s):**- a Linear Function Family (of linear functions).
- a Quadratic Function Family (of quadratic functions).
- a Cubic Function Family (of cubic functions).
- a Polynomial Function Family (of polynomial functions).
- an Exponential Function Family (of exponential functions).
- a Probability Function Family (of probability functions).
- …

**Counter-Example(s):****See:**Mathematical Analysis, Functional Analysis, Function Space, Harmonic Analysis, Group Representation.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/special_functions Retrieved:2015-6-14.
**Special functions**are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/list_of_mathematical_functions Retrieved:2015-6-14.
- In mathematics, many functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed out of statistics and mathematical physics. A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations.

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/list_of_mathematical_functions#Elementary_functions Retrieved:2015-6-14.
- Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...)

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/list_of_mathematical_functions#Algebraic_functions Retrieved:2015-6-14.
- Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
- Polynomials: Can be generated by addition, multiplication, and exponentiation alone.
- Constant function: polynomial of degree zero, graph is a horizontal straight line
- Linear function: First degree polynomial, graph is a straight line.
- Quadratic function: Second degree polynomial, graph is a parabola.
- Cubic function: Third degree polynomial.
- Quartic function: Fourth degree polynomial.
- Quintic function: Fifth degree polynomial.
- Sextic function: Sixth degree polynomial.

- Rational functions: A ratio of two polynomials.
- Nth root.
- Square root: Yields a number whose square is the given one [math]\displaystyle{ x^{\frac{1}{2}} \!\ }[/math] .
- Cube root: Yields a number whose cube is the given one [math]\displaystyle{ x^{\frac{1}{3}} \!\ }[/math] .

- Polynomials: Can be generated by addition, multiplication, and exponentiation alone.

- Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/list_of_mathematical_functions#Elementary_transcendental_functions Retrieved:2015-6-14.
- Transcendental functions are functions that are not algebraic.
- Exponential function: raises a fixed number to a variable power.
- Hyperbolic functions: formally similar to the trigonometric functions.
- Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.
- Power functions: raise a variable number to a fixed power; also known as Allometric functions; note: if the power is a rational number it is not strictly a transcendental function.
- Periodic functions.
- Trigonometric functions: sine, cosine, tangent, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.

- Transcendental functions are functions that are not algebraic.
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/list_of_mathematical_functions#Basic_special_functions Retrieved:2015-6-14.
- Indicator function: maps
*x*to either 1 or 0, depending on whether or not*x*belongs to some subset. - Step function: A finite linear combination of indicator functions of half-open intervals.
- Heaviside step function: 0 for negative arguments and 1 for positive arguments. The integral of the Dirac delta function.

- Sawtooth wave.
- Square wave.
- Triangle wave.
- Floor function: Largest integer less than or equal to a given number.
- Sign function: Returns only the sign of a number, as +1 or −1.
- Absolute value: distance to the origin (zero point)

- Indicator function: maps