Variables & the Function Machine

There is a mysterious machine on the desk: feed it 3, it hands back 7; feed it 5, it hands back 11. What rule is hiding inside? To crack the case you need two tools: letters that stand in for numbers, and functions that process them.

Letters stand in for numbers

Multiplication has an old rule: swap the two numbers and the result stays the same. Written with real numbers, $3 \times 5 = 5 \times 3$ describes exactly one pair. Written with letters:

$a \times b = b \times a$

One short line now covers every pair of numbers in the world. A letter that can hold any number like this is called a variable — an empty seat that any number may take.

Letters also change how multiplication is written: $3x$ is shorthand for $3 \times x$ , with the times sign left out.

Common trap

$3x$ means $3 \times x$ , not $3 + x$ . If $x = 4$ , then $3x = 12$ , not 7.

A function is a number machine

Drop a number into the machine, it applies a fixed rule, and a new number comes out. That is a function. Meet the machine whose rule is "double it, then add 1":

in 3: $3 \times 2 + 1 = 7$
in 5: $5 \times 2 + 1 = 11$

Mathematicians record the whole machine in one line:

$f(x) = 2x + 1$

Read it as "f of x equals 2x plus 1": $f$ names the rule, $x$ is the input waiting to go in, and $f(x)$ is the output that comes out.

Substitution: give the variable a job

Pick a value for $x$ and push it through the rule — that is called evaluating. Take $f(x) = 3x + 1$ :

If $x = 4$ : $f(4) = 3 \times 4 + 1 = 13$
If $x = 10$ : $f(10) = 3 \times 10 + 1 = 31$

Multiply first, then add. And notice the real payoff: the formula was written once, yet it handles every input. That is the whole point of variables — one rule covers every case.

The Guess-the-Rule Machine

The machine below is hiding a rule. Feed it inputs, watch the outputs, and hunt for the pattern until you can name $f(x)$ . Pulling one rule out of a pile of examples is exactly what algebra does all day.

InteractiveGuess-the-Rule Machine

A secret rule hides inside the machine. Feed it numbers and guess the rule!

? ? ?

Output

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Check yourself

Quick quiz

1. If f(x) = 2x + 3, what is f(5)?
2. What does 3x mean?
3. A machine turns 2 into 6 and 5 into 15. What is its rule?