Definitions - Functions

Function: Let X and Y two non-empty sets. A function (or map) is a rule or correspondence that associates each element of a set X with a unique element of another set Y. The set X is called the domanin and the set Y is called the codomain or image. One x € X cannot be associated with 2 or more elements of Y.  The fact that the correspondance goes from the from the domain to the codoimain is indicated by the notation f: XàY. We have a subset in set Y that is used in the correspondance. Ths subset is the range of the function.

Graph: The set of ordered pairs is the graph of f.

Ways to describe functions:

- list all function values

- list all pairs as ordered pairs

- list the elements of both the domain and codomain and connecting x values to function values by arrows

- in tabular

- in terms of x that calculates the value of f(x)

- graph

- describe in words

1-1correspondance: A correspondance between two sets in which each member of either set is paired with one and only one member of the other set.

Substitution value: The replacement of all occurances of a variable by quantity.

Equal functions: Two functions are said to be equal if and only if their domains are the same.

Origin: Lett here be given two perpendicular number lines: x axis and y axis. The intersection of them is called the origin of the coordinates system.

Quadrants: The axes divide the plane into quarters.

Linear function: A linear fucntion is a function f: RàR f(x)=ax+b where a and b are real numbers and a doesn’t equal 0. If a=0 we have a constant function. If b=0 we have direct proportionality relation.

Slope: The slope is the ratio of how much y changes per change in x.

Zero of a function: A zero of the function f(x)is a root of the equation f(x)=0, it is also the x-intersection.

Y-intersection: The value of f(0) is the value of the y-intersection of function f(x). You have to calculate the substitution value at 0.

Quadratic function: A quadratic function is a function F. RàR f(x)=ax²+bx+c where a, b and c are real numbers and a doesn’t equal 0. The graph of the function is called parabola.

Vertex: The point of intersection of the parabola with its axis of symmetry is called the vertex of the parabola.

Square-root function: A square-root function is a function f: ràR f(x)= √x

                                                                                                                                   x if x≥0

Absolute value function: An absolute value function is a function f: RàR f(x)= -x if x<0

Linear fractional function: A linear fractional function is a function f: R\{-d/c}àR f(x)= (ax+b)/(cx+d) where a,b,c and d are real numbers. The graph of the function is called hyperbola. 

                                                                                                                        1 if x>0

Signum function: The signum function is the function sgn(x): RàR f(x)=  0 if x=0

                                                                                                                      -1 if x<0

Integer part: The integer part of a real number is the largest integer n such that n≤x, it is denoted by: [x]

Fractional part: The fractional part of a real number is the difference x-n, it is denoted by {x}.

Properties of a function:

- domain

- peridodicity

- parity: even or odd

- boundary: bounded above or below

- monotony: increacing or decreasing

- zeros

- y-intersection

- extreme points: maximum and minimum

- range