https://file.aiquickdraw.com/mathbot/file/1729207057981pq2fo50k.jpg

To find the relationship between the input and output in the table, we need to identify a function that connects the two.

The given table shows:

\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{input} & 4 & 6 & 7 & n \\
\hline
\text{output} & 12 & 18 & 21 & ? \\
\hline
\end{array}
\]

### Step 1: Identify the pattern
Let's first check how the output changes as the input changes:

- From input 4 to 6: output changes from 12 to 18, an increase of \(18 - 12 = 6\).
- From input 6 to 7: output changes from 18 to 21, an increase of \(21 - 18 = 3\).

It seems the output increases by different amounts, so it may not be linear. Let's test whether a simple multiplication or addition rule applies.

### Step 2: Find a possible function
Let’s check if the outputs could be generated by multiplying or adding a constant. Checking a possible multiplication rule for input-output pairs:

- For input \(4\), output is \(12\): \(12 = 3 \times 4\).
- For input \(6\), output is \(18\): \(18 = 3 \times 6\).
- For input \(7\), output is \(21\): \(21 = 3 \times 7\).

It seems the rule is \( \text{output} = 3 \times \text{input} \).

### Step 3: Generalize the function
The function for this relation is:

\[
f(x) = 3x
\]

### Step 4: Calculate the output for input \(n\)
Given that the function is \(f(n) = 3n\), the output when the input is \(n\) is:

\[
\text{output} = 3n
\]

Would you like further details on this solution? Here are some related questions you might find useful:

1. What other methods could we use to find the function rule?
2. Can we confirm the function rule by plotting the points?
3. How would the function change if the increments in the output were not constant?
4. How do we find a function rule if it's not linear?
5. What is the significance of identifying functions from input-output tables?

**Tip:** Always check whether the pattern in the table is linear or nonlinear by observing the differences between consecutive outputs!

Write the output when the input is n, based on the table of input-output values: input {4, 6, 7, n}, output {12, 18, 21, ?}.

Learn how to determine the output of a function when the input is 'n' by identifying a linear pattern in the given input-output table: input {4, 6, 7, n}, output {12, 18, 21, ?}. The function is derived using multiplication, f(x) = 3x. Suitable for Grades 6-8.

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