Determine the unknown rule in a function machine with two operations, given specific input-output pairs.

The diagram shows an input, a first rule (unknown, marked with "?"), a second rule ($\times 7$), and a final output. We need to find the missing rule in the first box.
We have the following input-output pairs:

• Input 0, Output 21
• Input 2, Output (missing)
• Input 4, Output 49
• Input 5, Output 56
• Input 7, Output 70
• Input 9, Output 84

The second rule is "$\times 7$". To find the value before this rule was applied (the intermediate value), we divide the output by 7.

• For Input 0, Output 21: Intermediate Value $= \frac{21}{7} = 3$.
• For Input 4, Output 49: Intermediate Value $= \frac{49}{7} = 7$.
• For Input 5, Output 56: Intermediate Value $= \frac{56}{7} = 8$.
• For Input 7, Output 70: Intermediate Value $= \frac{70}{7} = 10$.
• For Input 9, Output 84: Intermediate Value $= \frac{84}{7} = 12$.

Now we look at the relationship between the original Input and the Intermediate Value:

• Input 0 $\to$ Intermediate Value 3
• Input 4 $\to$ Intermediate Value 7
• Input 5 $\to$ Intermediate Value 8
• Input 7 $\to$ Intermediate Value 10
• Input 9 $\to$ Intermediate Value 12 In each case, the Intermediate Value is 3 more than the Input.

$$0 + 3 = 3$$ $$4 + 3 = 7$$ $$5 + 3 = 8$$ $$7 + 3 = 10$$ $$9 + 3 = 12$$

Therefore, the missing rule in the first box is "$+3$".

Using the first rule ($+3$) and the second rule ($\times 7$) for Input 2:

• First, apply the "$+3$" rule: $2 + 3 = 5$.
• Next, apply the "$\times 7$" rule: $5 \times 7 = 35$. The missing output is 35.