NAME THE TOOL

A machine for numbers

Coach

Picture a machine. You drop a number in the top. The machine follows its rule, and one number drops out the bottom.

The panda's piggy bank works like this. Tell it a day, and it tells you how many coins are inside on that day. Day in, coins out.

Quick check: A machine's rule is "double it, then add 1." You drop in 5. What comes out?

THE SYMBOLS

Writing the machine down

We name a machine with a letter, usually f. The rule is written like this:

f(x) = 2x + 1

x is the input. 2x means 2 times x. f(3) means "feed 3 into f." The parentheses here do not mean times.

A rule can add a negative, like f(x) = 2x + (−1). Keep the negative in its parentheses.

Quick check: In f(4) = 9, which number is the input?

WORKED EXAMPLE

f(x) = 3x + (−2). What is f(4)?

Strategy: Swap x for the input, then run the rule

Name the rate or the pileTry a small caseDraw itAsk whyAnswer the exact thing asked

1

Understand

The input is 4. We want the output.

Why? f(4) asks what comes out when 4 goes in.

2

Plan

Put 4 everywhere x appears: 3 · 4 + (−2).

Why? x is just a holder for the input.

3

Solve

3 · 4 = 12. Then 12 + (−2) = 10.

Why? Multiply before you add.

4

Look back

Machine way: triple 4 to get 12, then add −2 to get 10. Same answer.

Why? Two ways agree, so 10 is safe.

Machine way

4 → triple → 12 → add −2 → 10

Symbol way

f(4) = 3 · 4 + (−2) = 10

Quick check: True or false: f(4) is 4, because 4 went in.

SEE IT

The machine, and two ways it jams

Trap: Doing the steps out of order

f(x) = 2x + 1 at 3. Adding first gives 2 · (3 + 1) = 8. The rule says double first: 6 + 1 = 7.

Trap: Reading 3x as 3 + x

f(x) = 3x at 4 is 3 times 4 = 12. It is not 3 + 4 = 7, and it is not 34.

Quick check: f(x) = 2x + 1. Which matches f(3)?

WHAT TO LOOK FOR

Clue words and easy mix-ups

f(3)Feed 3 into f. 3 is the input.
f(3) = 77 is the output. It is the answer to "what comes out?"
2x2 times x. Multiply before you add.
B(d) for the piggy bankB(4) = 10 means: on day 4 there are 10 coins. 10 is the pile, not coins per day.

Quick check: B(d) is the coins in the bank on day d. B(5) = 12. What is 12?

HOW THIS TRIPS PEOPLE

I fall for it first, so you don't

Coach

f(x) = 2x + (−1). Find f(6). I add first: 6 + (−1) = 5, then double: 10.

Coach

That ran the rule backward. Double first: 2 · 6 = 12. Then 12 + (−1) = 11.

Check habit: Multiply first, then add, and keep every negative in its parentheses.

Quick check: f(x) = 2x + (−1). Is f(6) equal to 11 or 13?

TRAP CHECK

Run this before every answer

Circle the input.
Swap x for it everywhere.
Multiply before you add. Keep negatives in parentheses.
Answer with the output, not the input.