A function machine represents a rule as a sequence of operations: an input goes in, is transformed step by step, and an output comes out. It's a visual bridge between arithmetic and algebra — the same machine that computes "input $\times 2$, then $+1$" is just a diagram for the formula $y = 2x+1$.

Apply each operation in order, left to right, using the result of one step as the input to the next.

Example: input $-4$, operations "$\times 2$" then "$+1$": $-4 \times 2 = -8$, then $-8+1=-7$.

Reverse both the order of operations and each operation itself (addition ↔ subtraction, multiplication ↔ division). The last operation applied going forward must be the first one undone.

Example: a machine applies "$\times 3$" then "$+15$", and the output is 48. Reverse order: undo $+15$ first ($48-15=33$), then undo $\times 3$ ($33 \div 3 = 11$). Input $= 11$.

Identify which operation touches the variable first. In $b = \dfrac{a-3}{5}$, the subtraction happens before the division (it's inside an implied bracket), so the machine is "subtract 3" → "divide by 5". A quick check: substitute a simple number for the variable and see which calculation you do first.

Draw the machine for the expression, then work backwards from the given output using inverse operations.

Example: solve $\dfrac{x}{2} - 6 = 3$. The machine is "÷2" then "−6", with output 3. Reverse: add 6 first ($3+6=9$), then multiply by 2 ($9\times2=18$). So $x=18$.

A machine applies "+5" then "×3" to an input, giving an output of 24. Find the input, then check your answer.

1. Reverse order, undo "×3" first: $24 \div 3 = 8$.
2. Undo "+5": $8 - 5 = 3$.
3. Check by running 3 forward through the original machine: $3+5=8$, then $8\times3=24$ ✓.

Answer: the input is 3. Always finish with this forward check — it catches order-of-operations errors before you commit to a final answer.