Function Notation
Function notation uses symbols like f(x) to represent the output of a function f when the input is x.
Formula
f(x)
Definition
Function notation is a shorthand way of writing a function: instead of "$y=$" we write "$f(x)=$," read "f of x," where $f$ is the function's name and $x$ is the input; other letters name other functions, like $g(x)$ or $h(t)$. More formally, it is a meta-linguistic convention mapping the name of a function and an argument to its value: in lambda calculus, $f(x)$ corresponds to applying the abstraction to an argument, and in formal logic, function symbols are part of the signature of a first-order theory.
Example
If $f(x) = 3x + 1$, then $f(2)$ means plug in $2$ for $x$: $f(2) = 3(2) + 1 = 7$. Given $f(x) = x^2 - 4$: $f(-3) = (-3)^2 - 4 = 5$, and $f(a + 1) = (a + 1)^2 - 4 = a^2 + 2a - 3$. In abstract algebra, a homomorphism $\phi: G \to H$ uses $\phi(g)$ to denote the image of group element $g$, and the notation generalizes further to functors $F(C)$ in category theory.
Key Insight
The parentheses in $f(x)$ do NOT mean multiplication, they mean "the input to function $f$ is $x$." This notation makes it easy to communicate, since $f(3)$ vs $g(3)$ clearly refer to different functions, and it enables clear expression of compositions like $f(g(x))$. Choosing meaningful function names (like $P$ for probability, $v$ for velocity) is a communication convention that aids comprehension in applied mathematics and physics.