While preparing a research station for departure, Nova and Luna discovered that the control panel rejected manual settings and responded only to a hidden function relating each input to an output. They recorded several input-output pairs, searched for the pattern, and tested a function rule that matched every observation. Before entering the final calibration values, they verified that the rule correctly predicted additional outputs. Once the system accepted the function, the station completed its safety checks, engines powered up, and the mission launched successfully.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Check how much the output increases when the input increases by 1.
Nova and Luna recorded the input-output pairs (1, 5), (2, 8), (3, 11), and (4, 14). Which function rule matches every pair?
f(x) = 3x + 2f(x) = 2x + 3f(x) = 3x - 2f(x) = x + 4Correct! The outputs increase by 3, and f(1) = 5, so the rule is f(x) = 3x + 2.
Try substituting more than one input into each rule and checking the outputs.
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Hint
Find f(6) first, and then substitute that result into the same function.
The calibration function is f(x) = 3x + 2. The system first uses input 6 and then uses the resulting output as the next input. What is the final output?
Correct! Applying the function twice gives a final output of 62.
Apply the function once to 6, then use that output as the new input.
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Hint
Substitute each recorded input and then test the new input 4.
Luna claims that the rule f(x) = x² + 1 matches the recorded pairs (1, 2), (2, 5), and (3, 10) and predicts an output of 17 when the input is 4. Is her calibration correct?
No, because f(2) should equal 4.No, because f(4) should equal 16.Yes, because the rule matches all three recorded pairs and f(4) = 17.Yes, but only because the outputs increase by the same amount.Correct! The rule matches every recorded pair, and f(4) = 4² + 1 = 17.
Check the rule by substituting the given inputs one at a time.
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🎉 Congrats! You finished this challenge set!
