The Definition of a Function

A function is simply a machine: you put a number in, and it hands exactly one number back. A vending machine is a good picture; press B4 and you get the same snack every time, never two different ones. The notation up top has three pieces: $f$ (the name of the machine), $x$ (what you put in) and $f(4)$ (what comes out when you put in 4). It looks like multiplication, but it isn't; it's just a label. Functions are how science writes "this depends on that."

Exactly one output. $y = x^2$ is a function: each $x$ gives one $y$. But $y^2 = x$ isn't, because $x = 4$ gives both $y = 2$ and $y = -2$. A machine that sometimes hands back two answers isn't much use.

Vertical line test. On a graph, the inputs run along the x-axis, so a vertical line picks out one input. If any vertical line crosses the graph twice, that input has two outputs, and it's not a function of $x$.

Evaluating. Replace every $x$ with the input, wrapped in parentheses so nothing gets lost: $f(a + 1) = (a + 1)^2 - 3(a + 1)$.

Domain and range. The domain is the inputs that make sense. Throw out anything that divides by zero or takes an even root of a negative number. The range is the outputs that actually come out.

A common mistake. It's tempting to think $f(x + h)$ is just $f(x) + h$, as if the notation distributed. It doesn't: $f(x + h)$ means "feed the whole thing $x + h$ into the rule." With $f(x) = x^2$, $f(x + h) = x^2 + 2xh + h^2$, which is not $x^2 + h$.

Every formula in science is a function: temperature as a function of time, dosage as a function of body mass, price as a function of demand. Spreadsheet cells and programming functions follow the same idea.