Let $f$ be the function... vs. Let $f(x)$ be the function...
When defining, or referring to, functions, I've seen both of the styles in the title. I was wondering which is considered to be more correct, or better style.
I've always found it strange to refer to $f(x)$ as a function, after all $f(x)$ is the value of the function at $x$. However, I suppose that we might agree that as long as $x$ is not declared anywhere $f(x)$ refers to the function itself, but as soon as $x$ is declared, then $f(x)$ is the value of the function at $x$.
Which do you prefer, and why? I am particularly interested in the context of undergraduate teaching, where the notation $f:X \rightarrow Y$ might be too foreign to be of use.
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$\begingroup$Whether to use $f$ or $f(x)$ for writing a functon, really depends on the context.
As you pointed out, $f(x)$ looks like the value of $f$ at $x$. But, the term $f(x)$ actually means to say "$f$ as a function of $x$". So, sually when defining it, it's more elegant to write it as $f:A\to B$ since you are explicitly mentioning the domain and range of $f$.
On the other hand, writing something like $f=x^2$ or $\int_a^b f\;\text{d}x$ looks weird, whereas $f(x)=x^2$ or $\int_a^b f(x)\;\text{d}x$ makes much more sense.
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