Domain and Ranges - Nerdstudy

every function can be expressed by graphing out its points which is something that we're probably already used to by now however we can also learn quite a bit about functions just by discussing its domain and range so in this video we're gonna explore more about what the domain and range of a function are and find the significance of both of them so before we begin this topic I want to emphasize that if you don't know the classification of numbers such as natural numbers whole numbers integers rational numbers irrational numbers and real numbers then we invite you to watch our classification of numbers video first before continuing on with this one otherwise let's get straight to it alright the domain of a function refers to all the possible X values that a function can take on that will yield a real y value so what do we mean by this well let's start with an example of a linear function and notice how this graph has arrows drawn on it and why because we can't quite draw this line in a way that extends forever so we draw the arrows to denote that it's supposed to continue on and on the graph certainly does not stop at these points because we can input any number that's ridiculously large or small and there will still be a multiplication of two and a subtraction of three applied to it to arrive upon a y value that is real so let's continue to think about what kind of number can be put into X would there be anything wrong with putting in a fraction into X definitely not putting in half would end up giving us 1 minus 3 which is negative 2 this is clearly a valid point on the graph so we know that X is not limited to only integers so then is X limited to only rational numbers or can we put in an irrational number and get a real number as our output well we certainly can if we plug in PI for X we would get 2 times pi minus 3 which will yield an irrational number that is roughly three point two eight three of course the digits and decimals just go on and on actually but we can still roughly plot that out onto the graph right about here notice how the continuous line just goes right through this point in fact since it's continuous you can imagine how many rational numbers in irrational numbers the functioned went through even just within the small segment that we can't even account for so since the x value can be inputted as both rational numbers or irrational numbers we can say that the x value can be any real number good so we've got ourselves a start to something called set builder notation for our domain in an effort to say that X can be any real number we can write this so let's just read this to make sense of it in English these braces basically mean the set of so the set of all X values such that X is an element which we can actually shorten by representing with this symbol here of all real numbers and there it is so to say something is an element of a real number is the same as saying that the number belongs to the category of real numbers so what this is basically saying is that our domain or all possible X values can be any number that is within the category of real numbers great so you'll notice that as we plug in a larger x value we also get a larger Y value and as we plug in a smaller x value we get a smaller Y value as well this brings us to the concept of range since just as X can be any rational number and irrational number making it any real number our Y or shall I say our range in this case can also be any real number as well to express this in set notation once again we can write it like so which stands for the set of all Y values such that Y is an element of all real numbers so it seems like for this linear function that there wasn't much limitation on what X can be and what Y can be but let's look at some more interesting scenarios so that we can get a good grasp of the different situations that we might see for both the domain and range let's take a look at this graph here it is just a vertical line the equation for this line is x equals 4 since regardless of what the Y values are the X is always going to be set at a value of 4 so in a situation like this we cannot just write that our domain is a set of all X values such that X is an element of all real numbers and why is this the case because notice here that it says an element of all real numbers well if we look at the graph here we see that X is only equal to 4 by saying all real numbers we would be implying five or six point five or even pie but none of these can be plugged into this graph because this graph only has 4 as its domain therefore we would instead write that our domain is a set of all X values such that X is equal to 4 fairly simple right oh and what would the range be well notice how the range of this graph or shall I say the Y values regardlessly continue to get larger or smaller so the range can be any number therefore for the range we can just write that it is in fact the set of all Y values such that Y is an element of all real numbers on a side note though if we want it to be more specific with a domain here we could actually add in the part where we mention that it is a set of all real numbers since we would just be specifying the classification of it but by further writing that X is only equal to 4 we are including a restriction within the classification therefore this would read the set of all X values such that X is an element of all real numbers where X is equal to 4 awesome so let's go ahead and try another example what about a parabola well here's the equation for this graph first of all notice how if we looked exclusively at the x-values for this graph the X can be any number from left to right this makes sense since there would be nothing wrong with squaring any x value that we plug in even zero can be squared since it would just be 0 so our domain for this equation would be the set of all X values such that X is an element of all real numbers now what about our range well here we have a more interesting case notice how our graph never goes below zero no matter what value it is in light of this if we look at the equation we realize that squaring any number would never result in a negative number except for zero which is neither a positive or negative number but we do know that our range at some point goes through every positive values such as integers like 1 2 3 and irrational numbers such as pi so between these two options which one would be the correct way to write the notation for the range here well this one would be the correct answer and although the first option might seem correct at a glance it is not quite accurate the second half of both options show that Y is greater or equal to 0 this is correct since the equation never yields any negative value no matter what x value we input so these parts of the two notations are correct in their restrictions the reason the first option is incorrect though is due to the first segment here this is because if we recall this notation denotes integers this part reads Y is an element of all integers by limiting the Y to only integers we are excluding for example any irrational numbers that the equation could possibly graph which would be totally incorrect since like we mentioned this graph does go through irrational numbers such as pi therefore the second option is the correct answer because Y is the element of all real numbers instead great now what might be a situation where we don't have real numbers being used in our domain or range because it seems to be the case that a lot of graphs do in fact use real numbers in their domain and range set notation well let's take a look at this graph noticing that this graph is not continuous like a line we would know right away that we are definitely not going to have real numbers in our domain and range set notation perhaps the graph is expressing a virtual reality simulation game let's say this game works in a way where the player finds himself or herself in a room and the room is however large that it needs to be there are always a number of people in the room and the person must write in the number of people generated in light of the domain how many people can the virtual reality game serve out well it can generate 0 people one person two three and so on perhaps even a thousand but definitely not half of a person nor can the game generate a negative number of people so we already have somewhat of a feel for the domain all right well one way to write the domain would be the following the domain is a set of all X values such that X is an element of all whole numbers and this is because if we can recall whole numbers are a set of numbers that start from zero and go on to positive infinity not including any decimal numbers or fractions so that would make sense for a domain here however it is a bit uncommon to write the domain as a set of whole numbers as people tend to use integers with some restrictions to describe the set in situations like this and like we already know integers include all the positive numbers from zero as well as the negative numbers so we would have to restrict all of them by including this part of this statement here where X is greater than or equal to zero so this is an example of a situation where we didn't use real numbers in our domain to describe it because if we had we would be implying that the game would sometimes serve out a randomly weird number of people like two point one two five seven people which certainly would not make sense now what could be right for arrange then well the minimum number of people the computer serves out is zero and if there are no other people in the room then only that one player would be in the room this means that the smallest possible range is one since one person is always in the room if the computer generates one person then there are two people in the room including the player so on and so forth again since the computer never serves 0.5 people in the room we can never get a fraction or a decimal like 1.5 as part of our range either or shall I say as part of our output so since we're talking about numbers starting from 1 that would go on to positive 2 3 4 and so on excluding fractions and decimals what kind of number category would be fitting for this well isn't it basically the natural numbers good so we can write our range as the range is the set of all Y values such that Y is an element of all natural numbers or remember like we mentioned earlier some people tend to like writing it as the classification of integers with some restrictions to it so instead we could write Y is an element of all integers where Y is greater than or equal to 1 so there it is the domain and range for this example awesome getting the hang of domain and range good so again domain and helps us to think about restrictions within our graph we can expect smooth lines that are continuous when we see real numbers or we can expect graphs to abruptly stop and not go past a certain point if we see extra conditions such as greater than or less than or greater than or equal to or less than or equal to signs so that was a long lesson but this is a very important concept that keeps recurring straight into University so where as some students never quite learn this well we encourage you to learn this as thoroughly as possible well thanks for sticking through with us until next time have a good one

Channel: Nerdstudy

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Keywords: Domain, Range, Nerdstudy, Domain and Range, Math, Real Numbers, Integers, Whole Numbers, Classification of Numbers, Range and Domain, Elements, Rational Number, Irrational Numbers, Natural Numbers

Id: UGXE8qqGS-Q

Length: 16min 24sec (984 seconds)

Published: Mon Mar 25 2019