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welcome to my presentation on domain of a function so what's the domain the domain of a function you'll often hear it combined with domain and range but the domain of a function is just what values can I put into a function and get a valid output so let's start with some example let's say let me make sure my PIN is all set up let's go ahead Oh wrong color good nice color color okay let's say I had f of X is equal to let's say x squared x squared so let me ask you a question what values of X can I put in here so I get a valid answer for x squared well I can put really put anything in here right any any real number so here I'll say that the domain domain is the set of X's such that X is a member of the real numbers this is just a fancy way of saying that okay this this are with this kind of double backbone here that just means real numbers and I think you're familiar with real numbers now that's pretty much every number outside of the complex numbers and if you don't know what complex numbers are that's fine you probably won't need to know it right now the real numbers are every number that most people are familiar with including irrational numbers including transcendental numbers including fractions every number is a real number so the domain here is X X just has to be a member of the real numbers and this little backwards looking e or something this means just means X is a member of the real numbers so let's do another one it's like variation somehow my circle tool got changed let's see ok so let's say I had f of X is equal to 1 over x squared so the same thing now can I still put any x value in here and get a reasonable answer well what's f of 0 something happened to my pen happen to my pen F of zero is equal to one over zero and what's one over zero I don't know what it is so so this is this is undefined undefined no one ever took the trouble to define what one over zero should be and they probably didn't do it so if some people probably thought about what it should be but they probably couldn't find out with a good definition for one over zero that's consistent with the rest of mathematics so one over zero stays undefined F of zero is undefined so we can't put zero in and get a valid answer for F of zero so here we say the domain domain is equal to dual brackets that shows kind of the set of what X's apply that's the little curly bracket I'd and draw out that well X is a member of the real numbers still such that X does not equal zero so here I just made a slight variation on what I had before before we said for when f of X is equal to x squared that X is just any real number now we're saying that X is any real number except for zero this is just a fancy way of saying and then these curly brackets just mean a set let's do a couple more ones let's say f of X is equal my pen is I have to find a better way let's see is equal to the square root of x minus three so just you know up here we said well this this function isn't defined when we get to zero in the denominator well what's interesting about this function can we take a square root of a negative number well until we learn about imaginary and complex numbers we can't so here we say well any X is valid here except for the X's that make this expression under the radical sign negative so we have to say that X minus 3 has to be greater than are equal to 0 right because you could have the square root of 0 that's fine this is zero so X it has to be greater than or equal X minus three has to be greater than or equal to zero so X has to be greater than or equal to three so here our domain is let me make sure I have enough space X is a member of the real numbers such that X is greater than or equal to three BAM let's do a slightly more difficult one what if I said f of X f of X is equal to the square root of absolute value of X minus three so now it's getting a little bit more complicated well just like this time around this expression under the radical still has to be greater than or equal to zero so you can just say that the absolute value of X minus 3 is greater than or equal to zero so we have the absolute value of x has to be greater than or equal to 3 and if order for the absolute value of something to be greater than or equal to something that that means that the net that that means that X has to be less than or equal to negative 3 or X has to be greater than or equal to 3 right it makes sense because X can't be negative 2 right because negative two has an absolute value less than 3 so X has to be less than negative 3 it has to be further in the negative direction than negative 3 or it has to be further in the positive direction than positive 3 so once again X has to be less than negative 3 or X has to be greater than 3 so we have our domain so we have it is X as a member of the reals so that I should always forget it such that the line I forget it's either : or a line I'm I'm rusty it's been years since I've done this kind of stuff but anyway I think you get the point it could be any real number here as long as X is less than negative 3 less than or equal to negative 3 or X is greater than or equal to 3 well let me ask a question now what if instead of this it was my pen my pen let's say instead of this we that was the denominator so let me this is all a separate problem up here so now we have 1 over the square root of the absolute value of X minus 3 so now how does this change the situation so now not only does this expression in the denominator not only know has this have to be greater than or equal to 0 can it be 0 anymore well no because then you would get the square root of 0 which is 0 and you get a 0 in the denominator so it's kind of like this problem plus this problem combined so now when you have 1 over the square root of the absolute value of X minus 3 now it's no longer greater than or equal to 0 it's just greater than 0 right it's just greater than 0 because we can't have a 0 here in the denominator so if it's greater than 0 then then we get rid of then we just say greater than 3 and essentially you just get rid of the equal signs right here let me let me erase it properly you get it's a slightly different color but maybe you won't notice so there you go so that's actually we stood another example since we have time let's see let me erase this okay now let's say that my pen is let's see oh I'm using the wrong color that's why let's say that F I don't oh yeah now I know let's say that f of X is equal to two if X is even and 1 over X minus 2 times X minus 1 if X is odd so what's the domain here what is a valid X I can put in here so immediately we have two clauses if X is even we use this clause so f of let's oh man my pen messed up again so let's say let's say that F of 4 well that's that's just equal to 2 because we use this clause here but this Clause applies when X is odd and just like we did in the last example what are the situations where this kind of breaks down well when the denominator is 0 well the denominator is 0 when X is equal to 2 or X is equal to 1 right but this clause only applies when X is odd so X is equal to 2 will apply to this clause so only X is equal to 1 would apply to this clause so we can only have so the domain is X its member of the reals such that X does not equal 1 I think that's all the time I have for now have fun practicing these domain problems