Domain and Range of Functions in Different Forms

hello in this video we’ll discuss about domain and range of a function as you know what is domain domain is set of all possible input elements and what is range yes range is set off all corresponding output elements right now here you see the word set of the word set is very very very important to understand what is the set set is well defined collection of objects this is a very general definition of sight in mathematics we normally consider these objects to be numbers and these numbers are actually the elements which we’re talking about okay well define well defined such a rule which defines which elements will be as the input elements for our domain so well defined is very important that demarcates whether this element will be in our to me or not okay so that’s what well defined is normally if we don’t define our set in that gate case what’s the default set default set for us is always X belongs to real numbers set of real numbers okay this symbol here is belongs to the Greek symbol epsilon so X belongs to set of real numbers I hope you remember real numbers in short real numbers what are you know so real numbers are all the numbers which could be represented on a number line okay square root 2 pi n number all these numbers are real numbers the numbers which cannot be represented on number line are complex numbers okay for example square root of a negative number it is not real okay so those are excluded but all others which can be represented belongs to the set of real numbers just a recap we have other sets like set of natural numbers natural numbers are numbers like 1 2 3 and so on some of whole numbers whole numbers include 0 and natural numbers don’t include 0 and then we have set of integers right sometimes they represent by Z sometimes by I so integers has a plus and minus natural numbers plus and minus members of liber8 okay and also zero okay so it’s a super set of whole numbers and natural numbers it includes both of them and also the negative numbers real numbers include fractions and square roots and all those things great so these are a set of numbers which we will consider now how to represent it to me that is the next main question for us to answer and then we’ll get into detail how to write the main or range of a function K now domain can be written in many different forms one of the most common form is the list form so let me just write down first the ways in which we can write one is list form this is also called the roster form and then we have interval notation I’ll talk about these just in a minute

okay and lastly we will consider the set builder form now list for list form is normally used wherever you have a list like tables right in tables we have a list where X values are given corresponding to Y values so better to write a list form for example let’s say we have this table where some X values are given and some y values of U and lesser the x values are 0 1 2 5 7 and 4 2 this is the value is 6 and 4 5 it is 7 4 9 it is 10 let us say this is devil is given to us where this is defined by a function G and x and y are so related that when you feed in you you get Y as an out in that case we can write domain in list form and that’s very simple just make this curly brackets curly brackets means set off okay and list all the values with comma inputs are 0 2 5 and 7 bracket closed so list off these X values may comes over to me and what’s the range range are the Y values just list them with comma in between 1 6 7 10 1 6 say one another place where list form is should be used is set of ordered pairs now let me give you a set of ordered pairs let’s say the set of ordered pairs here is 2 5 3 2 0 5 minus 1 6 set of ordered pairs they may not be in any particular order fine well the x-values are not repeating so it represents a function okay well you can write domain and range of a relation also right but we are at present just considering functions trying to focus on functions anyway now domain here is these x-values so I’ll just list them as 2 3 0 minus 1 and the range is the Y values we list them 5 2 5 5 v I’ve already written so will not repeat it ok we should not write it again in just 6 of course it is better when you make this kind of a list put them in order well it is advisable you should okay now here is a map integral let’s say in a mapping diagram we got these values 1 is that’s rated one minus one is also associated with minus 1 the arrow shows the association in this direction 2 with 4 0 0 now in this case they say this is our function in that case we can write domain of this function as so we’ll try to list them in a particular order so that looks nicer okay so minimum is minus 1 like minus 1 0 1 and 2 and the range here is 1 4 and 0 so let’s write them in order 0 1 & 4 well this is not wrong but this is better to write so in such cases we prefer to use list form okay now I’ll talk about interval notation interval notation is normally used when you have a continuous function where you want to represent sets in the form of real numbers okay where X belongs to real numbers for example let me just

draw a graphic let’s say this allows you that’s the FX or you can say the Y value and here is the function this is the x coordinate okay this is 1-1 let’s say we have a function here which is kind of going like this including this value and then we have a dysfunction on the side we’ll say including this and going parallel to this this is values minus 2 this is 1 for us ok no you can’t list all these values in between it’s kind of impossible right you don’t know what values are right so interval notation is very good for such situations we could have used set-builder for moles okay well now let’s see how to use interval notation to represent this kind of a function in interval notation we use brackets we first you should know that we are assuming that here X belongs to real numbers we don’t really write that X belongs to real number but it is taken as default in this we include two types of crackers this is the opening bracket which says start from here but do not include this becomes a closing bracket start from here but do not include but in case if you want to include then we should use these kinds of brackets these brackets means start from here and include okay we can also have combinations of these well there are no restrictions on that depending on our usage we can use as required in this particular function you can see that this function extends to left from minus 1 minus 1 it goes left how put the Y values of the function it is always minus 2 let’s consider the domain first right domain here is equal to it goes up to infinity can you ever reach infinity no we come so for infinity we’ll always use this bracket okay and it is approaching minus infinity so minus infinity and starts from minus 1 minus 1 is minus 1 included no it is not that is a whole whole represents that it is not included now our function starts then there is a break in between then starts a key from 1 onwards point 1 is included ok so that included will be shown by a square bracket right including 1 and then going forever that means infinitely and you cannot go to infinity so the domain is Union of poor things it includes both these x-values falling in this region so we can write Union okay this symbol U is Union at times we can also write calm all right and then just say this and this so this is one way of writing it this is interval notation very convenient to write now let me write domain not domain here’s the y-values so what are the permitted y-values for us one is minus two okay so write minus 2 and then what it is it is from here to here so you see there is one value minus 2 and then a region in such cases will prefer to use a set builder form ok so our next step is going to set builder form but this is a good place to start in set builder form as I said we’ll represent it as a set okay now here X belongs to real numbers rather y belongs to real numbers since we are talking about the range okay and then we’ll give condition this is so that or such that or beer what happens to y where y equals to minus 2 or what else or Y is greater than equal to 0 dragon roars so this is a set builder form where we have used these brackets to indicate a set and we have clearly specified that the range is a set of

real numbers where y value is minus 2 and then Y is greater than equal to zero okay so here we use combination of our interval and set builder form we’ll move forward and see some scatter plots and try to write their domain and range now I just hear some plot for you very simple from the graph as I told you if there is a scatter plot which is the best way of representing our domain and range well list form is pretty good so what is the domain view domain is the X values of these points so the x value of this point is 1 2 3 4 this is 4 right and this 3 this one – this one one hole – and here it is – so the domain is set of numbers from – 4 – 3 – 2 – 1 and how would range range is the Y values 1 2 this is minus 2 K and here plus 2 for these three for this and for for this so ranges minus 2 nothing on 1 2 3 & 4 easy so this form is pretty easy to write let’s move on to another example here again I have purposely given you a combination okay somewhere it is continuous somewhere it is not gaps in between remember if there are gaps in between good way to write it is in the set builder form or in interval notation this is also a continuous graph right so X belongs to real number so here will prefer to write in set builder form so say domain here is equal to X belongs to real numbers so that look at this is minus 2 is it ok anything less than minus 2 so we say and minus 2 is included so X is less than equal to minus 2 comma then we have from minus 1 to plus 1 but not including 1 and minus 1 so we have minus 1 X is greater than minus 1 but less than 1 and here this values 1 2 3 4 after 4 so X is greater than equal to 4 ok yes how about range range in this case can be given as everything from minus infinity to this place so as Y belongs to real numbers so that Y is less than equal to 2 this value is 2 or Y is equal to this value of 4 or Y is equal to this value of Phi so that is the way to give you range of this number of this function we will will take some more examples let us take equation okay so we have already seen how to write domain and range for a function if it is given in the form of a table or ordered pair or mapping diagram or a scatter plot now let us consider some equation okay fine let’s consider a situation where let’s say let us take an example like this let’s say some of some more to natural numbers is ten now how will you do this what is it domain and what is the range so natural numbers okay zero is not included remember that so how will you get ten well these are the combinations you can have one plus nine you’re gonna

have 2 + 8 3 + 7 4 + 6 5 Plus 5 6 plus 4 7 3 8 2 + 9 1 sorry so these combinations could give you set of natural numbers adding to 10 so of course the domain becomes these values so that’s the domain okay and was a dummy range ranges these values do you get it yes well let us twist this problem a bit and let’s remove the condition natural numbers okay now let’s consider example where let me change it in this form this is sum of two posit is ten now see when we say sum of two positive numbers then we mean that their real numbers and these real numbers are from I mean not negative because we have mentioned positive so this situation can be written as X plus y equals two okay where x and y are two real numbers okay when we write a equation in this particular form it is called implicit form let me write this work for you from this equation we can always write y equals to minus X plus 10 now if we can isolate y and write our equation in terms of y equals to something then that becomes explicit form why am i touching on this point it’s kind of important the spoilers has been missed earlier when we are talking about function notation then we said that in a equation y can be changed with FX right if that equation represents a function if it does not represent a function you cannot replace Y with FX but you know what it is not always very easy to explicitly write y as a function of X you know sometimes the values are so mixed up that we just can’t do it and if you cannot do it will not get explicit form and then we can’t write Y as FX now once I have written this in explicit form I can write this as FX equals 2 minus X plus 10 I hope this extra bit which you learn today you will appreciate it in times to come now y equals 2 minus X plus 10 what is this this is kind of the straight line but minus X plus 10 y intercept is 10 and a line goes down like this okay we are only considering a set of positive numbers right so in that case what happens this that this is our set of positive numbers whose sum is 10 okay now this is 10 and this is also 10 listen there could be many expertise in between if we add this x value and this Y value in general x and y we will get a sum of 10 so in this case what is it to me not domain here could be written set for or a little interval notation okay so we will write in short whenever you get an opportunity like this and freedom you should use the interval notation here the domain is from zero to ten okay okay I should say including boards right and how about range ranges also from zero to ten okay these are the Y values and this is X zero to ten you see when we changed our example from a set of natural numbers to positive real numbers if I don’t mention it means what real numbers don’t forget all this if I don’t mention then it means real numbers okay set of real numbers correct now we will see

some more equations rather today we’ll close here with the last example of a circle let’s say the equation of a circle as you know is x square plus y square equals to R square let’s take radius of five in that case X square plus y square is 25 okay let’s try to write this in explicit form in that case it becomes Y square equals 2 minus X square plus 25 and y equals 2 plus and minus square root of minus X square plus 25 since there are two values you know it is not a function right anyway let’s try to sketch it if we sketch it we get a circle you know circles not a function right it fails what is the line test here this point will be minus 5 this will be plus Phi plus five minus five because the radius is five is okay so that’s a circle what is the domain and range here domain is X belongs to real numbers where X is between minus 5 and plus 5 we could have written in interval notation also well let me write this in set builder form and range is y y belongs to real numbers where the value of y is between minus 5 and plus 5 okay it is not a function but we can always write its domain and range it does have a domain it’s a relation okay so today we learn about some very basic simple cases and we have understood that we can write domain and range in three different forms one of them is list for the other one interval notation set builder form you can also mix and match as per your convenience list form is very good in case you’re considering sets in the form of tables ordered pairs mapping diagrams or scatter plots interval form and the set builder form is normally used for equations and graphs where the things are continuous a continuous graph because you can’t really list all the real numbers in between two points there is a finite number of numbers between real numbers between any two points so they can’t resist it so we have this option of interval notation or set builder for beauty of set builder form is less versatile you see we can always write set of what and include different kinds of conditions as you have seen already there are very a lot of restrictions using interval notation as you’ve seen somewhere if something continuous comes and there is a break it may be difficult for us to use interval notation okay now word problems we can write an equation in case you can write an equation in the form of explicit form you can always replace Y as FX f of X and then sketch the graph get your function well in the next video on domain and range we will take scenarios for different algebraic equations for example polynomials reciprocal equations square root functions the basic functions and then see how we can write their domain and range okay I hope you learned from here and wait to see the next video we will discuss different kinds of algebraic equations okay thank you have a good day