The numbers are 7, 11, 15. The numbers are increasing by 4 each time. This means the rule involves multiplying the input position by 4. Let's test this for Input 1: 1 * 4 = 4. To get to the output of 7, we must add 3. So the rule is: Multiply the input position by 4, then add 3. Let's check for Input 2: 2 * 4 + 3 = 8 + 3 = 11. It works. Let's check for Input 3: 3 * 4 + 3 = 12 + 3 = 15. It works. Therefore, Choice B is correct. The other choices fail on at least one of the pairs.

First, find the rule. The input increases by 6 (16 - 10), and the output increases by 3 (10 - 7). This means the input is multiplied by 1/2 (or divided by 2). Let's test this. For an input of 10: 10 ÷ 2 = 5. To get to the output of 7, we must add 2. The rule is: Divide by 2, then add 2. Let's check with the second pair: 16 ÷ 2 + 2 = 8 + 2 = 10. It works. To find the input for an output of 13, we must reverse the rule: subtract 2, then multiply by 2. 13 - 2 = 11. 11 * 2 = 22. The input was 22.

First, find the rule. The number of gems increases by 2 (6 - 4), and the points increase by 10 (35 - 25). This means each gem is worth 5 points (10 ÷ 2 = 5). Let's test this. For 4 gems: 4 * 5 = 20. To get to 25 points, there must be a 5-point bonus. The rule is: Multiply the number of gems by 5, then add 5. Let's check with 6 gems: 6 * 5 + 5 = 30 + 5 = 35. It works. Now apply the rule to 10 gems: 10 * 5 + 5 = 50 + 5 = 55.

First, find the rule. The number of cookies increases by 3 (5 - 2), and the number of chips increases by 18 (33 - 15). This means each cookie gets 18 ÷ 3 = 6 chips. For 2 cookies, this is 2 * 6 = 12 chips. Since she uses 15 chips, there must be an extra 3 chips added to every batch. The rule is: Multiply the number of cookies by 6, then add 3. Let's check with 5 cookies: 5 * 6 + 3 = 30 + 3 = 33. It works. Now apply the rule to 4 cookies: 4 * 6 + 3 = 24 + 3 = 27.

First, find the rule. The input increases by 2 (from 3 to 5), and the output increases by 8 (from 13 to 21). This means the rule involves multiplying by 4 (8 ÷ 2 = 4). Let's test this. For an input of 3: 3 * 4 = 12. To get to the output of 13, we must add 1. So the rule is: Multiply by 4, then add 1. Let's check with the second pair: 5 * 4 + 1 = 20 + 1 = 21. It works. Now apply the rule to the new input, 8: 8 * 4 + 1 = 32 + 1 = 33.

When you see a temperature changing over time with a consistent pattern, you're dealing with a linear relationship. The key is finding how much the temperature changes per minute, then working backward to the starting point.

First, calculate the rate of change. From 2 minutes to 5 minutes (a 3-minute span), the temperature went from 19 degrees to 34 degrees. That's an increase of $$34 - 19 = 15$$ degrees over 3 minutes, so the rate is $$15 ÷ 3 = 5$$ degrees per minute.

Now work backward from any known point. Using the 2-minute mark: if the temperature was 19 degrees at 2 minutes, and it increases by 5 degrees each minute, then 2 minutes earlier (at 0 minutes) it would have been $$19 - (2 × 5) = 19 - 10 = 9$$ degrees.

Let's check why the other answers are wrong. Choice A (4 degrees) would mean the temperature increased by 15 degrees in just 2 minutes, giving a rate of 7.5 degrees per minute—but this doesn't match the 5-degree rate we calculated. Choice B (5 degrees) would create a rate of 7 degrees per minute from 0 to 2 minutes, again inconsistent with our pattern. Choice C (14 degrees) would mean only a 5-degree increase in 2 minutes (2.5 degrees per minute), which contradicts the 15-degree increase we observed from minutes 2 to 5.

For linear pattern problems, always find the rate first, then use it consistently to work forward or backward. Double-check by verifying your answer produces the same rate throughout the problem.

When you encounter a function machine problem, you need to find the mathematical rule that transforms each input into its corresponding output. Test each given rule against both data points to see which one works consistently.

Let's check answer choice D) Multiply the input by 2, then add 1. For the first pair (4, 9): $$4 \times 2 + 1 = 8 + 1 = 9$$ ✓. For the second pair (10, 21): $$10 \times 2 + 1 = 20 + 1 = 21$$ ✓. This rule works perfectly for both pairs.

Now let's see why the other options fail. Choice A) Add 5 to the input gives us $$4 + 5 = 9$$ (correct for the first pair) but $$10 + 5 = 15$$ (should be 21, so this fails). Choice B) Multiply by 3, then subtract 3 produces $$4 \times 3 - 3 = 12 - 3 = 9$$ (correct) but $$10 \times 3 - 3 = 30 - 3 = 27$$ (should be 21, so this fails). Choice C) Multiply by 1, then add 5 gives us $$4 \times 1 + 5 = 9$$ (correct) but $$10 \times 1 + 5 = 15$$ (should be 21, so this fails).

The key strategy for function machine problems is to always test your suspected rule against every given data point. A single correct result isn't enough—the rule must work for all pairs. Start by testing the simpler operations first, but don't assume the first rule that works for one pair is automatically correct.

Let's determine the rule. The input changes from 10 to 25 (an increase of 15), and the output changes from 4 to 19 (an increase of 15). This means the rule is simply to subtract a constant number. To get from 10 to 4, we subtract 6. The rule is: Subtract 6. Let's check: 25 - 6 = 19. It works. To find the input that gives an output of 34, we must reverse the rule, which means adding 6. 34 + 6 = 40. The input was 40.

To find the input when given the output, we must perform the inverse (opposite) operations in the reverse order. The reverse of 'subtract 5' is 'add 5'. The reverse of 'multiply by 3' is 'divide by 3'. So, we start with the output 28, add 5, and then divide by 3. 28 + 5 = 33. 33 ÷ 3 = 11. The input was 11. We can check this: (11 * 3) - 5 = 33 - 5 = 28.

First, find the rule. The input increases by 4 (7 - 3), and the output increases by 16 (27 - 11). This means the rule involves multiplying the input by 4 (16 ÷ 4 = 4). For an input of 3: 3 * 4 = 12. To get to the output of 11, we must subtract 1. The rule is: Multiply by 4, then subtract 1. Let's check with the second pair: 7 * 4 - 1 = 28 - 1 = 27. It works. Now, apply the rule to an input of 5: 5 * 4 - 1 = 20 - 1 = 19.