Turn on thread page Beta

Formal Definition of a Function watch

Announcements
    • Thread Starter
    Offline

    2
    ReputationRep:
    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!
    Offline

    14
    ReputationRep:
    In lecture notes, both typed and written, I usually see either:

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

    or

    

\begin{align*}

\\f:\mathbb{R} & \mapsto\mathbb{R}

\\ x & \mapsto e^{2x}

\end{align*}

    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.
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by FireGarden)
    In lecture notes, both typed and written, I usually see either:

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

    or

    

\begin{align*}

\\f:\mathbb{R} & \mapsto\mathbb{R}

\\ x & \mapsto e^{2x}

\end{align*}

    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.
    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?
    Offline

    18
    ReputationRep:
    (Original post by FireGarden)
    In lecture notes, both typed and written, I usually see either:

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

    or

    

\begin{align*}

\\f:\mathbb{R} & \mapsto\mathbb{R}

\\ x & \mapsto e^{2x}

\end{align*}

    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.
    Not quite. Whilst this is the conventional way to write a function, the domain of the OP's function is the positive reals so you cannot write f:\mathbb{R} \to \mathbb{R}; but rather f:\mathbb{R}_+\to \mathbb{R} (where you may define \mathbb{R}_+ = \{ x\in \mathbb{R} | x>0\}).

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

    (Original post by 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:
    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!
    The definition of "range" either refers to the image of f or a set containing the image of f (often referred to as the codomain) - and in most cases, depends on what question is being asked of you. The definition of domain is inflexible - it is simply the set of inputs for the function and must always be written in it's entirety - with nothing more or nothing less, if you intend to define the same function.

    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]).

    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.
    Offline

    14
    ReputationRep:
    (Original post by GingerCodeMan)
    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 format is (function name) : Domain -> Codomain, variable -> output.

    The domain should be precisely specified, if you want only positive reals, you'd usually write \mathbb{R}^{+}. 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.
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by Farhan.Hanif93)
    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. 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:
    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?
    Offline

    18
    ReputationRep:
    (Original post by 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.
    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.

    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?
    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.
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by 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 \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.
    Thanks. All of this makes much more sense now. I'm glad I know about this, but I'll probably stick to the simple method for the exam!
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: May 12, 2014

University open days

  • University of Derby
    Postgraduate and Professional Open Evening - Derby Campus Postgraduate
    Tue, 22 Jan '19
  • University of the West of England, Bristol
    Undergraduate Open Afternoon - Frenchay Campus Undergraduate
    Wed, 23 Jan '19
  • University of East London
    Postgraduate Open Evening Postgraduate
    Wed, 23 Jan '19
Poll
Brexit: Given the chance now, would you vote leave or remain?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Equations

Best calculators for A level Maths

Tips on which model to get

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.