# How to identify a function from ordered pairs

Graphs, Relations, Domain, and RangeThe rectangular coordinate systemA system with two number lines at right angles specifying points in a plane using ordered pairs (x, y). consist

## Graphs, Relations, Domain, and Range

The rectangular coordinate systemA system with two number lines at right angles specifying points in a plane using ordered pairs (x, y). consists of two real number lines that intersect at a right angle. The horizontal number line is called the x-axisThe horizontal number line used as reference in a rectangular coordinate system., and the vertical number line is called the y-axisThe vertical number line used as reference in a rectangular coordinate system.. These two number lines define a flat surface called a planeThe flat surface defined by x- and y-axes., and each point on this plane is associated with an ordered pairPairs (x, y) that identify position relative to the origin on a rectangular coordinate plane. of real numbers (x, y). The first number is called the x-coordinate, and the second number is called the y-coordinate. The intersection of the two axes is known as the originThe point where the x- and y-axes cross, denoted by (0, 0)., which corresponds to the point (0, 0).

The x- and y-axes break the plane into four regions called quadrantsThe four regions of a rectangular coordinate plane partly bounded by the x- and y-axes and numbered using the Roman numerals I, II, III, and IV., named using roman numerals I, II, III, and IV, as pictured. The ordered pair (x, y) represents the position of points relative to the origin. For example, the ordered pair (4, 3) represents the position 4 units to the left of the origin, and 3 units above in the second quadrant.

This system is often called the Cartesian coordinate systemTerm used in honor of René Descartes when referring to the rectangular coordinate system., named after the French mathematician René Descartes (15961650).

Figure 2.1

Rene Descartes Wikipedia

Next, we define a relationAny set of ordered pairs. as any set of ordered pairs. In the context of algebra, the relations of interest are sets of ordered pairs (x, y) in the rectangular coordinate plane. Typically, the coordinates are related by a rule expressed using an algebraic equation. For example, both the algebraic equations y=|x|2 and x=|y|+1 define relationsips between x and y. Following are some integers that satisfy both equations:

Here two relations consisting of seven ordered pair solutions are obtained:

y=|x|2 has solutions {(3,1),(2,0),(1,1),(0,2),(1,1),(2,0),(3,1)}andx=|y|+1 has solutions {(4,3),(3,2),(2,1),(1,0),(2,1),(3,2),(4,3)}

We can visually display any relation of this type on a coordinate plane by plotting the points.

The solution sets of each equation will form a relation consisting of infinitely many ordered pairs. We can use the given ordered pair solutions to estimate all of the other ordered pairs by drawing a line through the given points. Here we put an arrow on the ends of our lines to indicate that this set of ordered pairs continues without bounds.

The representation of a relation on a rectangular coordinate plane, as illustrated above, is called a graphA visual representation of a relation on a rectangular coordinate plane.. Any curve graphed on a rectangular coordinate plane represents a set of ordered pairs and thus defines a relation.

The set consisting of all of the first components of a relation, in this case the x-values, is called the domainThe set consisting of all of the first components of a relation. For relations consisting of points in the plane, the domain is the set of all x-values.. And the set consisting of all second components of a relation, in this case the y-values, is called the rangeThe set consisting of all of the second components of a relation. For relations consisting of points in the plane, the range is the set of all y-values. (or codomainUsed when referencing the range.). Often, we can determine the domain and range of a relation if we are given its graph.

Here we can see that the graph of y=|x|2 has a domain consisting of all real numbers, =(,), and a range of all y-values greater than or equal to 2, [2,). The domain of the graph of x=|y|+1 consists of all x-values greater than or equal to 1, [1,), and the range consists of all real numbers, =(,).

### Example 1

Determine the domain and range of the following relation:

Solution:

The minimum x-value represented on the graph is 8 all others are larger. Therefore, the domain consists of all x-values in the interval [8,). The minimum y-value represented on the graph is 0; thus, the range is [0,).

Answer: Domain: [8,); range: [0,)