# Formal Definition of a Function

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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:

where and ?

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:

Here's another attempt based on set-builder notation:

Any suggestions/examples would be greatly appreciated.

Thank you!

For example, how would you state the range and domain (formally) for the following function:

where and ?

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:

Here's another attempt based on set-builder notation:

Any suggestions/examples would be greatly appreciated.

Thank you!

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#2

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

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.

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.

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(Original post by

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

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.

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

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.

I think I understand . 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 ()? And what about the domain ()? Can you include inequalities on the left-hand side of the symbol?

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**FireGarden**)

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

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.

[I assume you just misread the OP's post, though]

(Original post by

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:

where and ?

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:

Here's another attempt based on set-builder notation:

Any suggestions/examples would be greatly appreciated.

Thank you!

**GingerCodeMan**)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:

where and ?

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:

Here's another attempt based on set-builder notation:

Any suggestions/examples would be greatly appreciated.

Thank you!

The more commonly used convention is one where "range = codomain", so that you can write, in general terms:

;

In this specific case:

;

(Where will do for the range, since ; and it's hopefully obvious what mean [if you have any doubt about the clarity of your domain or codomain, define the sets separately first]).

This is usually the assumed convention unless you're specifically

*asked*to give the range - in which case, the question is probably using the "range = image" convention, so there's little point writing out the entire function again; just write out the domain, image and nothing more.

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(Original post by

Sorry, I'm not sure I understand.

I think I understand . 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 ()? And what about the domain ()? Can you include inequalities on the left-hand side of the symbol?

**GingerCodeMan**)Sorry, I'm not sure I understand.

I think I understand . 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 ()? And what about the domain ()? Can you include inequalities on the left-hand side of the symbol?

The domain should be precisely specified, if you want only positive reals, you'd usually write . The range isn't usually specified, only the codomain. Of course, if it is easy to specify it, then you could write the range as the codomain, but it's often not a big deal - that's why people talk of surjectivity anyway, as the range and codomain may not be the same set.

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(Original post by

The more commonly used convention is one where "range = codomain", so that you can write, in general terms:

;

In this specific case:

;

(Where will do for the range, since ; and it's hopefully obvious what mean [if you have any doubt about the clarity of your domain or codomain, define the sets separately first]).

**Farhan.Hanif93**)The more commonly used convention is one where "range = codomain", so that you can write, in general terms:

;

In this specific case:

;

(Where will do for the range, since ; and it's hopefully obvious what mean [if you have any doubt about the clarity of your domain or codomain, define the sets separately first]).

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. Also, there's no real requirement to write them in this formal format; you can just write something like "Range: f(x) > ..., Domain: x ...". The reason I'm asking is just because I'm interested.

Just so I can solidify the concept in my head, here's another function:

Could be written as:

;

Is this correct?

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(Original post by

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.

**GingerCodeMan**)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.

*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 simply replaced with , 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:

;

will do as a definition for the function.

But only (or equivalents) is a suitable candidate for an explicitly-requested range.

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(Original post by

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

It's more common to see the simply replaced with , 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

So:

;

will do as a definition for the function.

But only (or equivalents) is a suitable candidate for an explicitly-requested range.

**Farhan.Hanif93**)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 simply replaced with , 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:

;

will do as a definition for the function.

But only (or equivalents) is a suitable candidate for an explicitly-requested range.

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