What does "$f: \mathbb{R} \to[0,\infty), f(x)=x^2$" mean? What is the co-domain, domain, or range? 
What does this mean?
  $$f: \mathbb{R} \to[0,\infty), f(x)=x^2$$
What is the co-domain, domain, or range here? I'm lost.

This might be a dumb question but I'm not able to read many questions properly due to this format. 
 A: The domain is $\Bbb R$.  The codomain $[0,\infty)$.  The range is $f(\Bbb R)$.
The range is used both for the image, as I have indicated here, and sometimes synonymously with codomain, or target space.
A: A function has three pieces of information: a domain (where the inputs come from), a codomain (where the outputs live) and a "rule", which is a recipe telling you how each input gets sent to its output.
Typically we write this as $f:A \to B$; here $f$ is the name of the function (the rule), $A$ is the domain and $B$ is called the codomain/target space.
The range/image of a function is slightly different from the codomain. The range of a function $f:A \to B$ is defined as the set of all possible outputs; or more formally $R_f = \{y \in B | \, \, \text{there exists an $x \in A$ such that $y = f(x)$}\}$. So, the range of a function is always a subset of the codomain, but not the other way around.
In your particular question, this is the exact same format. We have $f: \Bbb{R} \to [0, \infty)$, $f(x) = x^2$. Notice that we have given 3 pieces of information here:


*

*The domain of the function (the set of inputs/ "the thing before the arrow") is $\Bbb{R}$, which is the set of all real numbers.

*The codomain/target space is $[0, \infty)$, which is the set of non-negative real numbers.

*Lastly, we have the name of the function, which we call $f$, and the rule; $f(x) = x^2$. This means for any element $x$ in the domain of the function $f$, the output $f(x)$ is obtained by squaring that number.


Recall that we made a distinction between "codomain" and "range". In general, these two need not be the same sets, (the range is always a subset of codomain, but not necessarily equal to). But in this specific example, the range of $f$ and the codomain of $f$ are both equal to $[0, \infty)$.

Here's another example. Consider $\phi: \Bbb{R} \to \Bbb{R}$ defined by $\phi(x) = x^2$. 


*

*Here, the domain (the thing before arrow) is $\Bbb{R}$.

*The codomain (the thing after the arrow) is also $\Bbb{R}$. 

*The rule is the same as above (square the input). 

*The range of $\phi$ however, is $[0, \infty)$, which is different from the codomain.


So, although $f$ and $\phi$ have the same domain and same rule (and same range), we consider them to be different functions because they have different codomains of $[0, \infty)$ and $\Bbb{R}$ respectively.

Here's another example. Define $g: (0, \infty) \to \Bbb{R}$ by $g(x) = \dfrac{1}{\sqrt{x}}$. Then,


*

*The domain is $(0, \infty)$.

*THe codomain is $\Bbb{R}$.

*The rule for $g$ is as I have defined above.

*The range is $(0, \infty)$ (showing this requires a small calculation).


So, once again, this example shows that the range of a function need not be equal to the codomain.

Here's a simple/silly example to illustrate that the concept of functions is very general, and there is no need to even talk about numbers. The domain and codomain can be any sets. For example, let $E = \{\ddot{\smile}, \ddot{\frown}\}$ be the set of emotions, of happiness and sadness. Then, we can define a function $\xi: E \to E$ by
\begin{align}
\xi(\ddot{\smile}) = \ddot{\smile} \quad \text{and} \quad \xi(\ddot{\frown}) = \ddot{\smile}
\end{align}
In other words, every input gets sent to the happy output. So, here we have


*

*The domain is $E$, the set of emotions.

*The codomain is also $E$.

*The rule for $\xi$ is as I have defined above.

*The range of the function $\xi$ is $\{\ddot{\smile}\}$ "the set of happiness".


Hopefully these examples illustrate/clarify some of the terminology involved.
A: It means $f$ is a function that takes the entire real numbers (positive and negative) and maps it to the positive half line, and is of the functional form $x^2$.
Domain:  $(-\infty, +\infty)$
Range:  $[0, +\infty)$
A: The domain is the set of possible values for $x$. 
The codomain is the possible values for $f(x)$. 
Range is the subset of codomain which consists of the actual values that the function has mapped the domain to. 
So in this case, domain is $\Bbb R$, co-domain is as given in the question, real numbers greater than zero.
And the range is the same as co-domain because every positive real number has its square root.
A: *

*When a relation is a function, the expression " the $y$ that is related to $x$" becomes meaningful. The article " the" requires $(1)$ existence and $(2)$ unicity, like in " the husband of Ms Smith".


*It is "this  unique $y$ that is related to $x$" that is denoted by the symbolic expression : " $f(x)$" . ( Read : " the ( unique) image of $x$ under function $y$").


*This formula " the $y$ that is related to $x$" has the virtue of ( so to say) selecting ordered pairs (which never have the same second element if they have the same first element).
Note : An ordered pair is a pair with one element called " the first" one and another  element called " the second".

*

*Now,the crucial point is that the formula  " $f(x)=$ bla bla bla  " will select different sets of ordered pairs, depending on which set the first element comes from and wich set  the second element comes from.


*For example, if ( for each ordered pair that is to be selected) you only allow  the first element to come from set $A= \{ 1,2,3\}$ and the second element  to come from set $B=\{1,2,3,4,5,6\}$ , then, the formula "$f(x)=x^2$" will select the following pairs : (1,2) and (2,4) and your function will be reduced to this very modest collection  : $\{(1,2) , (2,4)\}$.


*Hence, which set plays the role of $A$ and of $B$ makes a noticeable difference. If you let the whole set of Real numbers play the role of $A$ (i.e. called the " domain") and the whole set of positive real numbers play the role of $B$, in that case, your formula will select a very different set of ordered pairs! Not only 2 ordered pairs, but infinitely many, and you will be able to represent this set of pairs as a set of points that make a continuous line.


*So, generally speaking , the expression :
$f: A \rightarrow B , f(x)=$ bla bla bla
reads as : the function $f$ from set $A$ ( domain) to set $B$, such that the image of $x$ under function $f$ is ( identical to) bla bla bla...

*

*The range of $f$ is the collection of all the images under $f$. This collection is necessarily a subset of set B ( possibly equal to B) .

Note : in terms of " ordered pairs" the range is the collection of all second elements.
