31.5 Stated Domain and Generated Range

Stated Domain and Generated Range is the process of finding a function's range when its domain is explicitly given as a finite list of allowed input values, by substituting each stated input into the function's rule using numerical function evaluation and collecting the resulting outputs into the range. Unlike finite range identification, which extracts range values directly from an already-complete relation, this process starts from only a rule and a restricted domain, requiring every output to be actively computed before the range can be stated.

This process connects function notation structure, numerical function evaluation, and finite range identification into a single combined procedure: the rule and notation define how to compute each output, evaluation carries out that computation for each stated input, and the resulting collection of outputs is treated exactly as a finite range would be, including the removal of any repeated values.

Explicit Allowed Input Set

Recognizing a Stated Domain

A stated domain is given directly as a set of specific input values that the function is restricted to, such as $$\text{domain}=\{0,1,2\}$$, rather than being left open to any number for which the rule happens to make sense.

Why the Domain Must Be Treated as a Restriction

Even if a function's rule could technically be evaluated at many other numbers, a stated domain restricts attention only to the listed values, meaning any input not appearing in the stated set is not considered part of this particular function's behavior.

Confirming the Domain Before Evaluation Begins

Before any computation starts, the stated domain is reviewed to confirm exactly which values must be substituted, since evaluating an input that was not actually included in the stated set would produce an output that does not belong to the function's range.

Domain Value Substitution Sequence

Establishing an Order for Substitution

Although the domain is a set without inherent order, working through its values in a consistent sequence, such as from smallest to largest, helps ensure that no stated input is skipped during the evaluation process.

Substituting Each Value in Turn

Each domain value is substituted into the function's rule one at a time, following the substitution and grouping practices described under numerical function evaluation, including grouping any negative or fractional inputs before simplifying.

Keeping Track of Completed Substitutions

As each domain value is evaluated, marking or recording it as completed helps confirm, once the full sequence has been worked through, that every stated input has actually been used to generate a corresponding output.

One Output Generated per Input

The Guarantee From the Function Definition

Because the object being evaluated is a function, each substituted input produces exactly one output, consistent with the function condition discussed under function classification verification, so the process of generating outputs never needs to consider the possibility of more than one result per input.

Computing the Single Output

For a given domain value, the single output is computed by fully simplifying the rule after substitution, following the post-substitution operation order described under numerical function evaluation until one numerical value remains.

Recording the Output Value

Once computed, the output value is recorded alongside the input value that produced it, preserving the pairing between the two even as the process moves on to the next domain value in the sequence.

Input-Output Pair Construction

Forming an Ordered Pair From Each Result

Each completed evaluation produces an ordered pair consisting of the domain value used and the output value obtained, written in the same form as any ordered pair discussed under relation representation recognition.

Building the Complete Set of Pairs

Repeating the pair construction step for every value in the stated domain produces a complete set of ordered pairs describing the function's behavior across its entire restricted domain.

Verifying the Pair Count Matches the Domain

The number of ordered pairs constructed should match the number of distinct values in the stated domain, since each domain value contributes exactly one pair, providing a simple check that no domain value was accidentally evaluated more than once or skipped entirely.

Generated Output Collection

Separating Outputs From the Constructed Pairs

Once all ordered pairs have been constructed, the output values are extracted from each pair, following the same second-coordinate extraction described under range entries from ordered pairs.

Assembling the Raw Output List

The extracted output values are assembled into a raw list in the same order the domain values were processed, preserving a clear connection back to which input produced which output before any further consolidation is applied.

Preparing for Duplicate Checking

With the raw output list assembled, the collection is ready to be checked for repeated values, since a stated domain with multiple distinct inputs can still produce a range with fewer distinct outputs if more than one input yields the same result.

Repeated Generated Output Removal

Why Duplicates Can Appear in Generated Outputs

Because a function is permitted to send different inputs to the same output, as discussed under shared output arrow convergence, the raw output list assembled from a stated domain can contain the same value more than once even though every individual evaluation was performed correctly.

Removing Duplicate Values From the Collection

Any output value appearing more than once in the raw list is reduced to a single entry, matching the same duplicate removal procedure applied when identifying a finite range directly from a given relation.

An Example of Removing Repeats in Generated Outputs

Evaluating $$f\left(x\right)=x^2$$ across the stated domain $$\{-2,2\}$$ produces outputs $$4$$ and $$4$$, which are reduced to the single distinct range value $$4$$.

Resulting Finite Range

Writing the Final Range Statement

After removing any repeated values, the remaining distinct outputs are written as a set enclosed in braces, forming the final range statement in the same format used for any finite range regardless of how it was obtained.

Confirming the Range Reflects Every Domain Value

A correctly generated range accounts for every value in the stated domain, meaning each distinct output in the final range set can be traced back to at least one domain value that produced it through the substitution process.

Using the Generated Range for Further Analysis

The resulting finite range, once obtained, can be used exactly as any other finite range would be, including comparison against a range obtained from a different representation of the same function as described under cross-representation function agreement, providing a check that the generated result is consistent and correct.