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f represents a function. What is the difference in meaning between f and f(x) ?

$f$ is the name of the function, while $f(x)$ represents the function value when the input is $x$.
For example, we says $f=\left\{ (x,y) \in \mathbb{R} \times \mathbb{R} \mid y=x^2+x \right\}$ or $f(x)=x^2+x$ when $x \in \mathbb{R}$