Does function have two meanings: both a relation and the function value?

Question

“In mathematics, a function1 is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x").” (wikipedia)

It seems like function has two meanings: (1) a relation between two variables, (2) the output of the function. Is this right? Or does f in ‘f of x’ only mean the output of a function, not just function?

Answer 1

The output or value of a function (at some input) is an element of a set. The function is a map $f: A \to B$, so for each $a \in A$ there's one and only one $b$ such that $f(a) = b$, ie. the value at input $a$ is always $b$. You can equivalently view a function as a type of relation $(a,b) \in f$ such that $(a,b), (a,c) \in f \implies b = c$. There is a bijection between sets satisfying respectively the two definitions.

Either way you view it is fine. Pick the one that's most convenient for the problem you're working with.