Domain is what goes in, range is what comes out. Domain is x, range is y. Domain is left to right, range is bottom to top! Have an example of where you're confused? More on reddit.com

A function is a device where you give it an object and it gives you some other object back. The collection of objects that you're allowed to give it, or the objects that it can handle, is called the domain. Anything that the function can accept as an input is a member of this collection. The type of objects that the function gives you back is called the codomain. You should think of codomain as a collection of objects that give the "form" that an output can take. Not every object in the codomain will necessarily be an output of the function, these are only the various outputs that you could get. The collection of outputs that you actually do get back for some input is called the range (or image) of the function. The range is therefore a subcollection of the codomain. You'll notice I haven't said anything about numbers so far, and that's because the idea of functions isn't limited to sets of numbers. If you've done any programming, you'll recall that functions can take any sort of input and give back any sort of output, like numbers, lists, tuples, other functions, or user created types. In the context of your typical algebra course, you'll look at functions like f(x) = 2/(x-1) + 3. Here, the implication is that the domain is some subset of the real numbers, and you are often asked to determine exactly what subset it is, i.e. determine which numbers the function cannot accept as an input. The codomain is also implied to be the real numbers, and so that range is some subset of the real numbers. Again, you're often asked to determine what subset this is. For the function f I wrote above, the domain would be all real numbers except 1, since using 1 as an input would create a division by 0. The range will be all real numbers except 3, since the fraction above will never be zero. Lastly, I'll stress that the typical algebra course's mentality of first giving you a function defined by a formula and then asking you to find the domain and codomain/range is often (but not always) backwards. In many cases, you'll want to create a function for which you know exactly what inputs and outputs the function will have, and then determine some logical rule for the function to follow. This is done in programming all the time. You have to know what inputs you're dealing with before you can decide what you're going to do with them. More on reddit.com

The [a, b] is closed interval notation. It tells us the endpoint a and b are closed. The { } is called set notation. You don’t really need to worry about it for now, but we can write [a,b] as the set. { x is real number | a <= x <= b} When infinity is in the domain, we can’t use closed interval. It has to be opened. And open interval is written as (a,b). We can have one endpoint be opened and the other be closed. It would be [a, b) or (b, a]. In your post it should be [0, infinity) The second notation tells us that the domain is all real numbers, excluding 0. Generally you should be allowed to just write that not using set notation. More on reddit.com

Well, you're talking about functions. A function takes some input and provides some output. The input could be anything, and the output could also be anything; it depends on the function. In math you use numbers, but you could use real life examples too. For example: let's say you go to an arcade. And let's say that at this arcade you can use nickels and dimes in the machines, but not pennies, and not quarters. And for every nickel you get 1 minute of play time and for every dime you get 3 minutes. Well in this silly example, the function is a translation of money -> time. In this function, the domain (allowed input) is a nickel or dime. And the range (possible output given the allowed input) is 1 or 3 minutes of play time. So even though a penny and a quarter is money, it's still not a part of the domain because the machines do not take that type of input. Likewise, there is no input that could ever provide exactly 2 minutes of playtime- its either 1 or 3 and that's it. Therefore, 2 minutes is not a part of the range. I suspect that when you say inequalities you're talking about "solving for x". Once you find x, what you're left with is literally the range, or rather, every valid output for a function. A very simple example is x > 3. This means every number larger than 3 is valid, but everything else is non-sensible (invalid), in the same way that the arcade machine example above you can never buy exactly 2 minutes of play time... it only supports 1 or 3 minutes; no other options. More on reddit.com