To determine the correct function that completes the function machine, we need to ensure we perform all specified transformations in sequence. Let's break down each step systematically:
1. First Transformation (multiply and subtract):
- Given input [tex]\( x \)[/tex], multiply by 6 and subtract 7:
[tex]\[
\text{Intermediate1} = 6x - 7
\][/tex]
2. Second Transformation (divide):
- Divide the result by 6:
[tex]\[
\text{Intermediate2} = \frac{6x - 7}{6}
\][/tex]
3. Third Transformation (add):
- Add 6 to the result:
[tex]\[
\text{Intermediate3} = \frac{6x - 7}{6} + 6
\][/tex]
4. Fourth Transformation (add):
- Add 7 to this result:
[tex]\[
\text{Intermediate4} = \left(\frac{6x - 7}{6} + 6\right) + 7
\][/tex]
5. Fifth Transformation (subtract):
- Subtract 6 from this result:
[tex]\[
\text{Intermediate5} = \left[\left(\frac{6x - 7}{6} + 6\right) + 7\right] - 6
\][/tex]
6. Final Transformation (multiply):
- Multiply this result by 6 to get the final output:
[tex]\[
\text{Output} = 6 \times \left\{\left[\left(\frac{6x - 7}{6} + 6\right) + 7\right] - 6\right\}
\][/tex]
These transformations yield specific intermediate results, but the critical detail here is ensuring we trace these operations accurately. Given the input [tex]\( x \)[/tex], the transformations align as follows:
[tex]\[
\boxed{6f(x) - 7}
\][/tex]
Thus, the function machine effectively implements these steps, and must preserve these transformations for the final multiplication to hold true.
Therefore, the function that completes the machine is represented accurately by adhering to:
[tex]\[
\boxed{f(x) = x}
\][/tex]
