32.6 Candidate Function Rule Verification

Candidate Function Rule Verification is the process of checking whether a proposed algebraic rule correctly matches a given table or set of ordered pairs, by substituting every listed input into the candidate rule and confirming that every predicted output agrees with the output actually recorded for that input. This process provides a disciplined way to test a guessed or proposed rule against real data, rather than accepting a rule as correct simply because it happens to work for one or two convenient input values.

Because a rule that fits some, but not all, of the given data is not a valid match for the entire table, this verification process requires checking every single row before a candidate rule can be accepted, treating the full set of given data as the standard the rule must satisfy completely.

Candidate Rule Statement

Proposing a Specific Rule to Test

Verification begins with a specific candidate rule stated explicitly, such as $$f\left(x\right)=3x-1$$, written in the same function notation structure used throughout function evaluation and representation work.

Where Candidate Rules Come From

A candidate rule can arise from noticing a pattern in a table's values, from a rule suggested in a problem, or from an educated guess based on the general shape of a graph, but regardless of its source, the rule must be stated precisely before testing can begin.

Treating the Candidate as an Unconfirmed Claim

Until it has been checked against every row of the available data, a candidate rule is treated only as an unconfirmed claim about the function, not yet accepted as an accurate description of the relationship the table or ordered pairs represent.

Table Input Substitution

Substituting Each Table Input Into the Candidate Rule

For every input listed in the table or ordered pair list being checked, that input is substituted into the candidate rule, following the same substitution and grouping practices described under input substitution into the function rule.

Working Through Every Row

Substitution is carried out for each row individually and in turn, ensuring that every input available in the data is eventually tested against the candidate rule rather than only a convenient subset.

Simplifying Each Substitution to a Predicted Value

After substitution, each resulting expression is simplified down to a single numerical value using the standard order of operations, producing what the candidate rule predicts the output should be for that particular input.

Predicted Output Comparison

Comparing the Predicted Value to the Recorded Value

Once a predicted output has been calculated for a given input, it is compared directly against the output actually recorded in the table or ordered pair for that same input, checking whether the two values match exactly.

Confirming an Agreement

If the predicted output matches the recorded output exactly, that particular row is considered consistent with the candidate rule, and verification proceeds to check the next row using the same comparison process.

Identifying a Disagreement

If the predicted output does not match the recorded output, that row reveals a disagreement between the candidate rule and the actual data, a finding that becomes central to the ultimate accept-or-reject decision about the candidate rule as a whole.

Every Table Row Requirement

Why a Single Match Is Not Sufficient

Because a candidate rule is meant to describe the entire relationship shown by the table, agreement at only one or a few rows does not establish that the rule is correct, since a rule could coincidentally match some inputs while failing to match others.

Checking the Complete Set of Rows

Verification requires substitution and comparison to be performed for every single row in the table or every single pair in the list, without skipping any row on the assumption that agreement elsewhere makes further checking unnecessary.

Stopping Early Only After a Disagreement Is Found

While every row should generally be checked, once even a single disagreement has been found, that alone is enough to reject the candidate rule, since a single mismatch already establishes that the rule does not correctly describe the entire table.

Matching Candidate Rule

Confirming Agreement Across All Rows

A candidate rule is confirmed as matching once every row of the table or every ordered pair in the list has been checked and found to agree between the predicted and recorded output values, with no disagreements found anywhere in the data.

Accepting the Rule as a Valid Description

Once full agreement has been confirmed, the candidate rule can be accepted as a valid algebraic description of the function shown by the table, meaning it can be used going forward to predict outputs for any input within the function's domain.

Communicating a Confirmed Match

A confirmed matching rule is typically reported alongside a brief summary of the checking process, such as noting that every row was tested and no disagreement was found, providing a clear record of how the confirmation was reached.

Rejected Candidate Rule

Recognizing a Rejected Rule

A candidate rule is rejected as soon as at least one row produces a predicted output that does not match its recorded output, regardless of how many other rows the rule happened to match correctly.

Reporting the Specific Point of Failure

A rejected candidate rule is reported together with the specific input at which the disagreement occurred and both the predicted and recorded output values, providing clear evidence for why the rule was rejected rather than a bare statement that it failed.

Moving Forward After Rejection

Once a candidate rule has been rejected, a new candidate rule must be proposed and subjected to the same complete verification process, since the rejected rule cannot simply be adjusted informally without being checked again in full against every row.

Finite Data Rule Nonuniqueness

More Than One Rule Can Match the Same Finite Data

Because a table or ordered pair list only ever specifies a function's behavior at a finite set of inputs, more than one distinct algebraic rule can sometimes agree with all of the given data, differing only in what they would predict for inputs not included in the table at all.

Why This Nonuniqueness Does Not Undermine Verification

Even though more than one rule might match the same finite data, the verification process itself remains valid: each candidate rule is checked independently against the given rows, and a rule that passes every check has been correctly confirmed as consistent with the available data, regardless of whether another equally consistent rule might also exist.

Being Cautious About Extending Beyond the Given Data

Because of this nonuniqueness, a confirmed matching rule should be understood as verified specifically for the inputs actually present in the table, and using that rule to predict outputs for entirely new inputs beyond the original data carries the implicit assumption that the same rule continues to apply beyond what was directly checked.