What does $f(\cdot)$ mean in math

Question

Let $f:\mathbb{R} \to \mathbb{R}$ be a function, what does $f(\cdot)$ mean usually? Is it another way of writing this function, or is it a real number?

Answer 1

It is another way of writing the function, emphasising that the value of $f$ at, for instance, $5$ is written as $f(5)$, and not $f5$ or $5f$ or $(5)f$ or $f|_5$ or anything else. The dot is just a placeholder.

Some would write this as $f(x)$ instead of $f(\cdot)$, but this is a slightly different emphasis again. The notation $f(x)$ tends to be associated to a specific description of $f$, for instance $f(x) = 4x-3$.

Answer 2

The meaning of the symbol $f(\,)$ can be somewhat ambiguous depending upon the context.

For example, if $f(x)=x^3$ then in the equation $y=f(x)$ the meaning of $f(x)$ is $x^3$.

However, in the equation $y=f(2)$ the meaning of $f(2)$ is the number obtained by cubing 2.

So in the former equation, it represents a mathematical expression and in the latter it represents a number.