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CCSS.Math:

we're asked to determine for each x-value whether it is in the domain of F or not and they have our definition of f of X up here so pause this video and see if you can work through this before we do it together all right so just as a bit of a review if X is in the domain of our function that means that if we input our X into our function then we are going to get a legitimate output f of X but for whatever reason F isn't defined at X or if it gets some kind of undefined state well then it is then X would not be in the domain so let's try these different values is X equal negative 5 in the domain of F let's see what happens if we try to evaluate F of negative 5 well then we in the numerator we get negative 5 plus 5 every place where you see an X we replace it with a negative 5 so it's negative 5 plus 5 over negative 5 minus 3 which is equal to in our numerator we get zero and our denominator we get negative 8 now at first you see the zero you might be a little bit worried but it's just a zero in the numerator so this whole thing just evaluates to a zero which is a completely legitimate output so x equals negative five is in the domain what about x equals zero is that in the domain pause the video see if you can figure that out well F of zero is going to be equal to in our numerator we have zero plus five in our denominator we have zero minus three well that's just going to get us five in the numerator and negative three in the denominator this would just be negative five thirds but this is a completely legitimate output so the function is defined at x equals zero so it's in the domain for sure now what about x equals three pause the video try to figure that out well I'll do that up here F of three is going to be equal to what and you might already see some warning signs as to what's going to happen here in the denominator but I'll just evaluate the whole thing in the numerator we get three plus five in the denominator we get three minus three so this is going to be equal to 8 over zero now what is eight divided by zero well we don't know this is one of those fascinating things in mathematics we haven't defined what happens when something is divided by 0 so 3 is not in the domain the function is not defined they're not in domain let's do another example determine for each x value whether it is the note whether it is in the domain of G or not so pause this video and try to work through all three of these alright so first of all when x equals negative 3 do we get a legitimate G of X so let's see G of negative 3 if we try to evaluate that's going to be the square root of 3 times negative 3 which is equal to the square root of negative 9 well we with just a principal square root like this we don't know how to evaluate this so this is not in the domain what about when x equals 0 well G of 0 is going to be equal to the square root of 3 times 0 which is equal to the square root of 0 which is equal to 0 so that gave us a legitimate result so that is in the domain now what about G of 2 or x equals 2 what does that give us a legitimate G of 2 well G of 2 is going to be equal to the square root of 3 times 2 which is equal to the square root of 6 which is a legitimate output so x equals 2 is in the domain let's do one last example so we're told this is H of X right over here and once again we have to figure out whether these X values are in the domain or not pause this video and see if you can work through that all right well let's just first think about H of negative 1 what's that going to be equal to negative 1 every place we see an X we're going to replace it with a negative 1 minus 5 squared well this is going to be equal to negative 6 squared negative 6 squared which is equal to positive 36 which is a very legitimate output and so this is definitely in the domain what about 5 so H of 5 is going to be equal to 5 minus 5 squared now you might be getting worried because you're seeing a 0 here not like we're trying to divide by zero' we're just squaring zero which is completely legitimate so zero squared is just a zero and so H of five is very much defined so this is in the domain now what about H of ten well H of ten is going to be equal to 10 minus five squared which is equal to five squared which is equal to 25 once again it's a very legitimate output so the function is definitely defined for x equals ten and we're done