Formal Definition of a Function

In lecture notes, both typed and written, I usually see either:

$f:\mathbb{R}\mapsto\mathbb{R}, x\mapsto e^{2x}$

or



edit: I have simply no idea why the second one isn't formatting properly.. hm. Well anyway, it's supposed to have x mapping to e^2x underneath the R's, looking a bit more neat than the one above, but a bit impractical in the layout sometimes.

In lecture notes, both typed and written, I usually see either:

$f:\mathbb{R}\mapsto\mathbb{R}, x\mapsto e^{2x}$

or



edit: I have simply no idea why the second one isn't formatting properly.. hm. Well anyway, it's supposed to have x mapping to e^2x underneath the R's, looking a bit more neat than the one above, but a bit impractical in the layout sometimes.

Hi, I was just wondering if there is a convention for writing the domain and range of a function?

For example, how would you state the range and domain (formally) for the following function:
$f : x \mapsto e^{2x}$ where $x \in \mathbb{R}$ and $x > 0$ ?

As far as I'm aware, to state the domain of the function, you can just leave a space after the function definition and then list the criteria, but I'm not sure if there's a nice way to write the range. According to this, you can write Dom(f(x)) = ... and Ran(f(x)) = ..., but I feel a bit iffy about this.

Here's one attempt at writing the domain and range of the function I mentioned:
$f : x \mapsto e^{2x}\,\,\,\,\,\, x \in \mathbb{R}, x > 0$
$f(x) \in \mathbb{R}, f(x) > 1$

Here's another attempt based on set-builder notation:
$f : x \mapsto e^{2x}\,\,\,\,\,\, \{x\, |\, x \in \mathbb{R} \land x > 0\}$
$\text{Ran}(f(x)) = \{x\, |\, x \in \mathbb{R} \land x > 1\}$

Any suggestions/examples would be greatly appreciated.

Thank you!

Sorry, I'm not sure I understand.
I think I understand $f : \mathbb{R} \mapsto \mathbb{R}, x \mapsto e^{2x}$. So does this mean that 'input a real number, output a real number', and then 'the output should be e to the power of 2 multiplied by the input'?

But what about defining the range ($f(x) \in \mathbb{R}, f(x) > 1$)? And what about the domain ($x \in \mathbb{R}, x > 0$)? Can you include inequalities on the left-hand side of the $\mapsto$ symbol?

The more commonly used convention is one where "range = codomain", so that you can write, in general terms:
$f:\{\text{Domain}\} \to \{\text{image and "other stuff"}\}$;
$x\mapsto f(x)$

In this specific case:
$f:\mathbb{R}_+ \to \mathbb{R}$;
$x\mapsto e^{2x}$

(Where $\mathbb{R}$ will do for the range, since $\mathbb{R}_{>1} \subset \mathbb{R}$; and it's hopefully obvious what $\mathbb{R}_+, \mathbb{R}_{>1}$ mean [if you have any doubt about the clarity of your domain or codomain, define the sets separately first]).

Thanks!

The 'range' must refer to the 'image' for my exam board, because the mark scheme gives the restrictive answers, i.e. not just 'the real numbers', but something like f(x) > 5/2 or something like that.

Just so I can solidify the concept in my head, here's another function:
$f : x \mapsto \ln{x}\,\,\,\,\,\,\, x \in \mathbb{R}, x > e^3$

Could be written as:
$\mathbb{R}_{>y} = \{x \in \mathbb{R} | x > y\}$

$f : \mathbb{R}_{(>e^3)} \mapsto \mathbb{R}_{(>3)}$;
$x \mapsto \ln{x}$

Is this correct?

Yeah, as above, if the questions asks for the range, you can give it in whichever form you like, provided that it's the image. So things like f(x) > 5/2 are perfectly valid.


It's correct. But again, as above, we conventionally refer to the range as a codomain when we use this format, so that it doesn't have to be the image (and more importantly, there's no implicit suggestion that the codomain is the image here [without specifying more about the function i.e. surjectivity] so there little point in writing the image as the codomain here as it's not explicit that f's image is being read off!)

It's more common to see the $\mathbb{R}_{>3}$ simply replaced with $\mathbb{R}$, as both are suitable codomains; and since it's often easier to see that a function has a real output than it is to compute it's image, it's often sensible to stop once you've determined a codomain.

So:

$f:\mathbb{R}_{>e^3} \to \mathbb{R}$;
$x\mapsto \ln x$

will do as a definition for the function.

But only $\mathbb{R}_{>3}$ (or equivalents) is a suitable candidate for an explicitly-requested range.