How $f:[a,b]\rightarrow[c,d]$ should be read?

Question

I found it in a book but I don't know what the ":" means. What does this expression mean?

It should simply mean a function, with the preimage [a,b] and image [c,d]. An other example would be $f:\mathbb{R}\to\mathbb{R}$ — Mr.Topology, Jun 26 '16 at 01:28

score 3 · Accepted Answer · edited Jun 26 '16 at 02:11

It means that it is a function that maps the numbers in the interval $[a,b]$ to numbers in the interval $[c,d]$. $[a,b]$ is the domain and $[c,d]$ is the codomain.

$f: X \rightarrow Y$ is fairly standard notation.

Example if $f(x) = x^2 - 7$ for $-1 \le x \le 5$ then $f:[-1,5] \rightarrow [-7,18]$ because $f$ maps the numbers in the interval $[-1,5]$ to the interval $[-7,18]$.

edited codomain.

score 1 · Answer 2 · answered Jun 26 '16 at 01:28

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It means that $f$ is a function whose domain is the closed interval from $a$ to $b$ and whose codomain is the closed interval from $c$ to $d$.

answered Jun 26 '16 at 01:28

Deusovi

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@Vitor: The codomain can be larger than the image. The image is just the set of things you get when you put everything from your domain through the function. The codomain is just a set that the image is a part of. – Deusovi Jun 26 '16 at 01:36
a codomain is the set which might e mapped to. The image is the set that is mapped to. Image $\subset$ codomain. Example if $f(x) = x^2 + 3x - 5$ we will say.. "that's a real function; it maps real number to real numbers-- the range/codomain is the real numbers". With bit of work and effort we can notice that actually the image is ... well, I'm too lazy to work it out... and who cares? We write $f:\mathbb R \rightarrow \mathbb R$ even though $f(x)$ is never actually less than negative whatever. – fleablood Jun 26 '16 at 01:40
@Deusovi Thank you. I think I understood. if the image is {1,2.5,3} the codomain is [1,3] – Vitor Abella Jun 26 '16 at 01:51
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@Vitor: Not necessarily. {1,2.5,3} can also be a codomain. You just need to give some codomain when you write down a function. It can be the same as the image, or it can be a superset of the image. – Deusovi Jun 26 '16 at 01:52

score 1 · Answer 3 · answered Jun 26 '16 at 01:28

We read $f: [a,b] \rightarrow [c,d]$ as "$f$ is a function from the closed interval $[a,b]$ to the closed interval $[c,d]$".

Alternatively, "$f$ maps the closed interval $[a,b]$ to the closed interval $[c,d]$".

In general, $f: A \rightarrow B$ means that $f$ is a function that maps elements from its domain $A$ to elements in its range $B$.

How $f:[a,b]\rightarrow[c,d]$ should be read?

3 Answers3