Function Notation
Functions & Advanced AlgebraFunction 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 writing "$y =$", we write "$f(x) =$" which is read as "f of x." The $f$ is the function's name, and $x$ is the input.
Example
If $f(x) = 3x + 1$, then $f(2)$ means "plug in $2$ for $x$." So $f(2) = 3(2) + 1 = 7$. The notation tells you exactly what to substitute.
Key Insight
The parentheses in $f(x)$ do NOT mean multiplication. They mean "the input to function $f$ is $x$." You could name the function $g$, $h$, or any letter.
Definition
Function notation $f(x)$ specifies the function name $f$ and the independent variable $x$. It emphasizes that the output depends on the input. Other letters are used for different functions: $g(x)$, $h(t)$, etc.
Example
Given $f(x) = x^2 - 4$, evaluate $f(-3)$: $f(-3) = (-3)^2 - 4 = 9 - 4 = 5$. Evaluate $f(a + 1)$: $f(a + 1) = (a + 1)^2 - 4 = a^2 + 2a + 1 - 4 = a^2 + 2a - 3$.
Key Insight
Function notation makes it easy to communicate: $f(3)$ vs $g(3)$ clearly refer to different functions. It also enables clear expression of compositions: $f(g(x))$.
Definition
Function notation 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. In formal logic, function symbols are part of the signature of a first-order theory.
Example
In abstract algebra, a homomorphism $\phi: G \to H$ uses $\phi(g)$ to denote the image of group element $g$. The notation generalizes to functors $F(C)$ in category theory.
Key Insight
Choosing meaningful function names (like $P$ for probability, $v$ for velocity) is a communication convention that aids comprehension in applied mathematics and physics.