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Worked example: domain of algebraic functions

CCSS.Math:

Video transcript

let's do some more examples finding domains of functions so let's add the function G of X so this is our function definition here tells us look if we have a input X the output G of X is going to be equal to 1 over the square root of 6 - let me write this a little bit neater 1 over the square root of 6 minus the absolute value of X so like always pause this video and see if you can figure out what what is the domain of this function based on this function definition what is the domain of G what is the set of all inputs for which this function is defined alright so to think about all of the inputs that would allow this function to be defined it might be easy to think well when does this function not get defined well if we're divided by 0 then we're not going to be defined or if we have a negative under the radical so if you think about it if if what we have under the radical is 0 or negative if it's 0 or negative it's 0 you can take the you can take the principal root of 0 it's going to be 0 but then you're going to divide by 0 and that's going to be undefined and if what you have under the radical is negative then the principal root isn't defined for a negative number or at least the classic principal root isn't defined for a negative number so if 6 minus the absolute value of x is 0 or negative this thing isn't going to be defined or another way to think about it is it's going to be defined so g-wiz is defined is defined if G is defined if 6 minus and maybe I could write if and only if sometimes people write if and only if with two F's right there if G is defined if and only if is kind of a Matthew way of saying if and only if 6 minus the absolute value of x is greater than 0 it has to be it has to be positive if it's zero we're going to take the square root of zero is zero and you're going to divide by zero that's undefined and if it's less than zero then you're taking your trying find the principal root of a negative number that's not defined so let's see it's G is defined if and only if this is true and let's see we could add the absolute value of X to both sides we could add the absolute value of x to both sides and then that would give us 6 is greater than the absolute value of X or that the absolute value of X is less than 6 or we could say that you know let me write it that way the absolute value of x is less than 6 or another way of saying that is X would have to be less than 6 and greater than negative 6 or X is between negative 6 and 6 these two things these two things are equivalent if the magnitude of X is less than 6 then X is X is greater than negative 6 and less than positive 6 so if we wanted to write the domain and kind of fancy domain set notation we could write the domain of G is going to be X all the X's that are a member of real numbers such that negative 6 is less than X which is less than 6 and we're done now let's do another one and this one's going to get even a little bit a little bit hairier just for just for kicks all right so let's say that I have let's say that I have H of X is equal to and I'm going to have kind of a composite definition here so let's say it's X plus 10 over X plus 10 times X minus 9 times X minus 5 times X minus 5 and it's this if it H of X is this if X does not equal 5 and it's equal to it's equal to PI if X is equal if X is equal to 5 so once again pause this video think about what is the domain of H or another way to think about when is H not defined so let's think about it so what would make H not defined so if some if X is anything other than 5 we go to this clause if it's five we go to this clause so in this clause up here what would make this thing undefined well the most obvious thing is if we divide by zero so what's going to cause us to divide by zero so if if X is equal to if let me let me write it here so we're going to divide by zero divided by zero that would happen if X is equal to nine that would happen if X is equal to negative 10 X is equal to negative 10 now we have to be careful would that happen if this was the only definition here it would happen when X is equal to 5 but remember what x equals 5 we don't look at this part of the compound definition we look at this part so it's true that up here you would be dividing by 0 of X equaling 5 but X equaling 5 you would never even look at it look there if our input is 5 you use this part of the definition so you would divide by 0 maybe I should write it this way divided by 0 on I guess you could say the top the top Clause or the top part of definition part of the definition if x equals 9 x equals negative 10 or and that's it because x equals 5 doesn't apply to this top part if this clause wasn't here then yes you would write x equals 5 now we're almost done but some of you might say wait wait wait but look can't I simplify this I have X plus 10 in the numerator X plus 10 in the denominator can I just simplify this and then that'll disappear and you could except if you did that you are now creating a different function definition because if you just simplify this in his you just said 1 over X minus 9 a 1 over X minus 9 times X minus 5 this is now a different function that one actually would be defined at x equals negative 10 but the one that we have started with this one is not this is you're going to end up with 0 over 0 you're to end up with that indeterminate form so for this function exactly the way it's written it's not going to be defined when X is equal to 9 or X is equal to negative 10 so once again if we want to fight it in our fancy domain set notation the domain is going to be X all the X's that are a member of the reals that X does not equal 9 and X does not equal negative 10 any other real number X it's going to work including 5 if x equals 5 H of H of 5 is going to be equal to PI because you default to this one over here H of 5 you say ok x equals 5 we do that one right over there now if you gave x equals 9 you're going to divide by 0 x equals negative 10 you're going to divide by 0 what's going to work for anything else so that right over there is the domain