Functions

A relation between two variables $x$ and $y$, is any set of points which are on the $(x, y)$ plane.

There exist four different types of relations:

• One to one

• One to many

• Many to one

• Many to many

A function is defined as a mapping onto a single value. Therefore one to one, and many to one relations are considered to be functions.

Testing if a relation is a function

1. Algebraic Method: If we substitute any value of $x$ and it results in a singular $y$-value, then it can be considered a function.
2. Graphical Method (Vertical Line Test): If we are given a plot of a function on the $(x, y)$ plane, where the $x$-axis is parallel with the horizontal, and the $y$-axis is parallel with the vertical, then if we are able to draw a vertical line anywhere on the graph, and it only intersects the plot once, it is a function. If the vertical line intersects the plot more than once, then it cannot be considered a function.

Domain and Range

Linear functions

Linear functions are functions which can be written in the form:

\[\boxed{y=mx+c} \quad \text{OR} \quad \boxed{y=m(x-x_1)+y_1}\]

Quadratic functions

Line of symmetry

For functions in the form $f(x)=ax^2+bx+c$, the axis of symmetry of is

\[\boxed{x=-\frac{b}{2a}}\]

Discriminant

\[\boxed{\Delta = b^2 -4ac}\]

The Quadratic formula

The solutions to $ax^2 + bx + c = 0$ are:

\[\boxed{x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}} \quad \text{where } a \neq 0\]