Function types

Procedures versus Functions:

A procedure is a description of a computation that, given an input, produces an output.

Functions or computational problems don't indicate how to compute the output from the given input.

Many methods might exist for the same specification.

A computational problem may allow several possible outputs for each input.

We will write procedures in Breeze; often, these are called functions, but we will reserve that term for the mathematical objects.

• One to One function:

f : D -> F is one-to-one if f (x) = f (y) implies x = y; that is, both x and y are in D.

• Onto function:

F: D -> F is onto if for every z element of F, there exists an element a in D such that f (a) = z.

A function is invertible if it is one-to-one and onto.

• Linear functions: Linear functions are the functions whose graph is a straight line. A linear function has the form z = f(x) = a + bx. A linear function has one dependent variable, and one independent variable. The dependent variable is z, and the independent variable is x.

• Polynomial function: A polynomial function involves only non-negative integer powers of x such as a quadratic, a cubic, a quartic, and so on. We can give a general definition of a polynomial, and define its degree. A polynomial of degree n is a function of the form f(x) = anx n + an−1x n−1 + . . . + a2x 2 + a1x + a0, where a's are real numbers, also known as coefficients of polynomials.

For example: f(x) = 4x 3 − 3x 2 + 2

• Identity function: For any domain D, idD: D -> D maps each domain element d to itself.

• Constant function: this is a special function represented as a horizontal line.

• Probability distribution function : This used to define the relative likelihood of different outcomes of a particular experiment. It assigns a probability to each potential outcome. Probabilities of all outcomes must sum equal to 1. Often, probability distribution is a uniform distribution. That means, it assigns the same probability to each outcome. When we roll a die, the possible outcomes are 1, 2, 3, 4, 5, and probabilities are defined as Pr(1) = Pr(2) = Pr(3) = Pr(4) = Pr(5) = 1/5.

• Gaussian Function: When the number of events is large, then the Gaussian function can be used to describe events. Gaussian distribution is described as a continuous function, also known as normal distribution. Normal distribution has the mean equal to median, and has symmetry about the center.