A relation is a set of ordered pairs.
Any set of ordered pairs automatically relates the set of first coordinates to the set of second coordinates. The domain of a relation is the set of all first coordinates. The range of a relation is the set of all second coordinates.
The graph of a relation involving real numbers is the collection of all ordered pairs plotted in the rectangular (Cartesian) coordinate system.
A function is a relation in which every element of the domain is paired with exactly one element of the range.
Equivalently, a function is a relation in which no two distinct ordered pairs have the same first coordinates. (2nd coordinates can be reused, but 1st coordinates cannot!)
Functions (and relations) are typically defined in one of four ways:
Vertical Line Test: If a relation can be graphed in the rectangular coordinate system, the relation is a function if and only if no vertical line passes through the graph more than once.
If even one vertical line intersects the graph two or more times, the graph is not the graph of a function.
In this course, the most common way in which a function will be defined is by a formula or equation.
Consider the function .
This function can be described in an equivalent way by the formula: .
In this context, the free input variable is the independent variable and the output variable is the dependent variable.
In functional notation, we name the function (rather than the dependent variable) and describe the relation: .
is the value of the function named when evaluated at the input value . It is often read " of " or " at ."
VERY IMPORTANT IDEA: When the domain of a function is not explicitly stated, we assume the domain is the largest set of real numbers that produce valid, real outputs from the function's definition. In this case, the domain is sometimes called the implied domain.
Consider the relation whose graph is shown below. Let's call it .