31.4 Finite Range Identification

Finite Range Identification is the process of listing every distinct output value that belongs to a relation or function when that relation is defined by a limited, countable set of pairs rather than by a rule covering an unbroken range of numbers. Because a finite relation only ever involves a specific, fixed collection of outputs, its range can be found simply by gathering every output value that actually appears across the relation's given pairs, mirroring the process used for finite domain identification but applied to the second coordinate of each pair instead of the first.

This identification process applies across every representation discussed under relation representation recognition, since ordered pair lists, tables, mapping diagrams, and graphs made of separate points all encode the same underlying set of output values, just displayed in different visual or symbolic forms.

Range Entries from Ordered Pairs

Reading Outputs Directly from Each Pair

When a relation is given as a list of ordered pairs, each pair's second coordinate is a member of the range, so scanning through the list and recording every second coordinate produces the complete collection of range entries before any further processing.

An Example of Extracting Range Entries

From the ordered pair list $$\{\left(1,4\right),\left(2,5\right),\left(3,6\right)\}$$, the range entries extracted are $$4$$, $$5$$, and $$6$$, taken directly from the second position of each listed pair.

Ignoring the First Coordinate During This Step

While extracting range entries, the first coordinate of each pair is set aside entirely, since it belongs to the domain rather than the range, and mixing the two would produce an incorrect list of output values.

Range Entries from Table Outputs

Reading Outputs from the Output Column or Row

When a relation is given as an input-output table, the range entries are the values listed in the column or row designated for outputs, read directly without needing to reference the paired input values at all.

An Example of Extracting Range Entries From a Table

A table listing output values $$5$$, $$7$$, and $$9$$ alongside their corresponding inputs has a range consisting of exactly those three listed output values, regardless of what the associated inputs happen to be.

Handling Tables With Non-Sequential Output Values

Table outputs are not required to follow any particular order or pattern, so range identification from a table involves reading every listed output value exactly as given, even if the values do not increase steadily or follow an obvious numerical sequence.

Range Entries from Mapping Targets

Reading Outputs from the Output Oval

When a relation is given as a mapping diagram, the range entries are the values listed inside the output oval that have at least one arrow pointing to them from the input oval, corresponding to every value actually produced by the relation.

Distinguishing True Range Entries From Unused Values

If the output oval of a mapping diagram happens to display a value with no arrow arriving at it, that value is not actually part of the relation and is excluded from the range, since range membership requires participation in at least one ordered pair.

An Example of Extracting Range Entries From a Mapping Diagram

Range Entries from Finite Graph Points

Reading Outputs from Plotted Points

When a relation is given as a graph consisting of a finite collection of separate, isolated points rather than a continuous curve, the range entries are the vertical coordinates of each plotted point, read directly from the graph's axis.

Locating Vertical Positions Precisely

Because range entries from a graph depend on accurately reading each point's vertical position, care must be taken to align each point with the correct axis value, particularly when points are plotted close together or the axis scale involves values other than whole numbers.

An Example of Extracting Range Entries From Plotted Points

Points plotted at vertical positions $$2$$, $$6$$, and $$10$$ on a coordinate grid contribute exactly those three values to the range, regardless of their individual horizontal positions.

Repeated Range Value Removal

Why Repeats Can Occur During Extraction

Because a single output value can appear paired with more than one input, particularly for relations where several inputs share a common output as described under shared output arrow convergence, the raw extraction process described in the previous sections can produce a list containing the same range value more than once.

Removing Duplicate Entries

Once every output value has been extracted from the given representation, any value that appears more than once in the raw list is reduced to a single entry, since the range is defined as a set of distinct values rather than a list that tracks repetition.

An Example of Removing Repeats

Extracting outputs from the ordered pair list $$\{\left(1,4\right),\left(2,4\right),\left(3,9\right)\}$$ initially produces the raw list $$4$$, $$4$$, $$9$$, which is reduced to the final range entries $$4$$ and $$9$$ after removing the repeated value.

Finite Range Set Statement

Writing the Range as a Set

Once all range entries have been extracted and any repeats removed, the range is written as a set enclosed in braces, listing each distinct output value exactly once, commonly arranged in increasing numerical order for clarity even though set membership does not require any particular order.

An Example of a Completed Range Statement

The range extracted from the earlier ordered pair example is written as $$\text{range}=\{4,9\}$$, presenting the final result as a clean, unambiguous set of values.

Using the Range Statement Alongside the Domain

A correctly identified finite range, paired with the finite domain identified from the same relation, together provide a complete summary of every input and output value involved, forming the basis for fully describing the relation's behavior across its entire finite set of ordered pairs.