This is simply a basic notation question: what is the meaning of $$f:\mathbb R \rightarrow \mathbb R$$ I imagine it's some sort of function to do with the set of real numbers, perhaps some sort of mapping. Until now I've only encountered functions of the form $$f(x)=...$$ or $$f:x\mapsto...$$ Thanks in advance.
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asked Aug 6, 2020 at 20:13
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1$\begingroup$ You can read this as "$f$ is a function that maps from the reals to the reals." It doesn't tell you anything else about what that function is. $\endgroup$– Brian TungCommented Aug 6, 2020 at 20:14
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$\begingroup$ It looks as if you are already aware of the 2 ways to define a function. First we define the domain and co-domain ($f:\mathbb R\to \mathbb R$). Then we define the actual mapping with either $f(x)=\ldots$ or $f:x\mapsto\ldots$. Sometimes the definition of domain and co-domain is left out if it is considered obvious from the context. $\endgroup$– Klaas van AarsenCommented Aug 6, 2020 at 20:44
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$\begingroup$ @arjun Not range, but codomain. $\endgroup$– MJDCommented Aug 6, 2020 at 20:47
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1$\begingroup$ The term "range" is considered ambiguous as it could either be the codomain or the image of the domain. Best to avoid it. $\endgroup$– Klaas van AarsenCommented Aug 6, 2020 at 20:49
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$\begingroup$ Refer also to What is a function? $\endgroup$– userCommented Aug 6, 2020 at 21:24
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2 Answers
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It means that the function maps any real number (all real numbers are the domain) to one and only one real number (the codomain is a subset of $\mathbb R$).
For example the following notations are valid
- $f:\mathbb R \rightarrow \mathbb R,\: f(x)=x^3$
- $f:\mathbb R \rightarrow \mathbb R,\: f(x)=x^2$
- $f:\mathbb R \rightarrow \mathbb R^+_0,\: f(x)=x^2$
- $f:\mathbb R^+_0 \rightarrow \mathbb R^+_0,\: f(x)=\sqrt x$
and the following are wrong
- $f:\mathbb R \rightarrow \mathbb R^+_0,\: f(x)=\sqrt x$
- $f:\mathbb R \rightarrow \mathbb R,\: f(x)=\log x$
- $f:\mathbb R \rightarrow \mathbb R,\: f(x)=\frac 1x$
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$\begingroup$ Given a function, is it possible to find the codomain? For example, can you find the codomain of $f(x)=x^2$? $\endgroup$ Commented Aug 7, 2020 at 9:57
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$\begingroup$ @A-levelStudent Yes of course, in this case since $x^2\ge 0$ we have that the codomain is any $U\subseteq \mathbb R$ such that $\mathbb R^+_0\subseteq U$. $\endgroup$– userCommented Aug 7, 2020 at 10:28
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It means that $f$ is a function with domain $\Bbb R$ and codomain $\Bbb R$.
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$\begingroup$ Just to doublecheck: codomain is simply the range? $\endgroup$ Commented Aug 6, 2020 at 20:17
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$\begingroup$ All the values are reals. But it is possible that some reals are not values. Some texts use "range" for the set of values of $f$. Thus the use of "codomain" where the set of values might or might not be all of $\mathbb R$. $\endgroup$– GEdgarCommented Aug 6, 2020 at 20:18
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1$\begingroup$ @BrianTung Often Codomain and Range are used as synonyms and Image is used to indicate the set of all $f(x)$ with $x$ in the domain. $\endgroup$– userCommented Aug 6, 2020 at 20:26
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1$\begingroup$ @user: I know. That's why I only said that "range" is "often used more specifically..." :-) $\endgroup$ Commented Aug 6, 2020 at 20:35
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1$\begingroup$ @A-levelStudent: It means that every value the function takes on (as output) falls within the reals. If you're familiar with function declarations in computing languages, it's like declaring that a function takes a float as input, and produces a float as output (except of course that reals are not floats). That declaration does not means that every float can be produced as an output; it just says that all outputs are floats. $\endgroup$ Commented Aug 6, 2020 at 21:07