Function notation writes a function as f(x): f names the function, x is the input, and f(x) is the output it produces. Learn to read f(x) correctly, evaluate a function at a given input by substitution, and see why function notation makes naming and transforming functions clearer than writing y.
What function notation is
Function notation is the way we write a function using its name, like f(x), instead of the more informal "y = ...". Function notation names the rule (f), shows the input (x) in parentheses, and treats the whole expression f(x) as the output — the value the function produces for that input.
f(x): f is the function's name, x is the input, and f(x) is the output it produces.
Reading f(x)
The expression "f(x)" is read "f of x" — NOT "f times x". The parentheses do not mean multiplication here; they hold the input value. This distinction matters because function notation and multiplication look similar but mean completely different things.
Evaluating a function
To evaluate a function at a specific input, replace every x in the rule with that input's value. For f(x) = 2x + 1, evaluating at x = 3 means: f(3) = 2(3) + 1 = 7. The number in the parentheses (3) plugs directly into the rule wherever x appears, and this works the same way whether the function comes from an equation, a table, or a graph.
Why function notation is useful
Function notation lets you name and reuse multiple functions clearly — f(x), g(x), and h(x) can all exist in the same problem without confusion, unlike using "y" for everything. It also makes operations like dividing functions or checking whether a function is one-to-one much easier to express precisely, since you can write exactly which function and which input you mean.
Function notation with transformations
Function notation also describes transformations directly: f(x) + 2 shifts the graph up 2 units, while f(x + 2) shifts it left 2 units. Reading these transformations correctly — and not mixing up a shift inside the parentheses with one outside — depends entirely on understanding what f(x) means in the first place.