An input-output table shows: input 3 → output 10, input 5 → output 16, input 7 → output 22. What is the rule?
AMultiply by 3, then add 1
BAdd 7
CMultiply by 2, then add 4
DMultiply by 5, then subtract 5
Test 'multiply by 3, then add 1': 3 × 3 + 1 = 10 ✓; 5 × 3 + 1 = 16 ✓; 7 × 3 + 1 = 22 ✓. Rule confirmed. 'Add 7' works for the first pair (3 + 7 = 10) but fails for the second (5 + 7 = 12, not 16). This is exactly why you must test at least two pairs before trusting a rule — one matching pair is often coincidental.
Question 2 Multiple Choice
A table shows: input 2 → output 10, input 4 → output 20, input 6 → output 30. A student says the rule is 'add 8.' Why is this wrong?
AThe rule is 'add 10' — the student made an arithmetic error
BThe rule is 'multiply by 5' — the student only checked one pair and missed the multiplicative pattern
CThe student is right — adding 8 to each input gives the correct outputs
DThe rule is 'add 8 then add 2' — a two-step additive rule
'Add 8' seems to work for the first pair (2 + 8 = 10), but fails immediately for the second (4 + 8 = 12, not 20). The correct rule is 'multiply by 5.' The common error is looking only at the difference between input and output for one pair, rather than checking consistency and testing multiplicative relationships. Always check the ratio (output ÷ input) as well as the difference (output − input).
Question 3 True / False
When finding the rule for an input-output table, testing your guess against at least two input-output pairs is necessary to confirm the rule.
TTrue
FFalse
Answer: True
One pair is never enough. Many different rules can match a single pair: for input 3, output 10, both 'add 7' and 'multiply by 3 then add 1' work. A second pair eliminates most false candidates. A rule must hold for every pair in the table without exception. Testing two or more pairs dramatically increases confidence and catches rules that only coincidentally match the first pair.
Question 4 True / False
If the rule for an input-output table is 'multiply by 4,' then an input of 6 could produce different outputs on different tables using the same rule.
TTrue
FFalse
Answer: False
This is the fundamental property of a function (and an input-output table): a consistent rule maps each input to exactly one output, always. If the rule is 'multiply by 4,' input 6 always produces output 24 — no exceptions, no variation. An input-output 'rule' that sometimes gives different outputs isn't a rule at all. This consistency is what makes the rule useful and what distinguishes a function from a random pairing.
Question 5 Short Answer
An input-output table shows: input 3 → output 11, input 6 → output 20. A student guesses the rule is 'multiply by 3, then add 2.' Verify whether this is correct.
Think about your answer, then reveal below.
Model answer: Test pair 1: 3 × 3 + 2 = 11 ✓. Test pair 2: 6 × 3 + 2 = 20 ✓. Both pairs confirm the rule. 'Multiply by 3, then add 2' is correct.
Always substitute both known pairs. A single pair could fit many rules; multiple pairs narrow it to one consistent rule. Notice that input 3 → output 11 could also fit 'add 8,' but 'add 8' fails for input 6 (6 + 8 = 14, not 20). The two-step rule 'multiply by 3 then add 2' is the only simple rule that satisfies both pairs — and that is your confirmation.