## Answer :

The rule for this function machine is described by the equation:

[tex]\[ \text{Output} = 6x + 47 \][/tex]

Let's break down the equation:

1.

**Input**: The input to the function machine is denoted by [tex]\(x\)[/tex].

2.

**Processing Rule**: Each input [tex]\(x\)[/tex] is multiplied by 6.

3.

**Addition**: After multiplying by 6, we add 47 to the result.

Therefore, the equation that represents the function machine is:

[tex]\[ \text{Output} = 6x + 47 \][/tex]

To visualize this with examples:

- If the input [tex]\(x = 1\)[/tex]:

[tex]\[ \text{Output} = 6(1) + 47 = 6 + 47 = 53 \][/tex]

- If the input [tex]\(x = 2\)[/tex]:

[tex]\[ \text{Output} = 6(2) + 47 = 12 + 47 = 59 \][/tex]

- If the input [tex]\(x = 5\)[/tex]:

[tex]\[ \text{Output} = 6(5) + 47 = 30 + 47 = 77 \][/tex]

Thus, the general form of the equation for the function machine is:

[tex]\[ \boxed{6x + 47} \][/tex]

This equation succinctly captures the relationship between the input [tex]\(x\)[/tex] and the output produced by the function machine.