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CCSS.Math:

we're told this table defines function f alright for every X they give us the corresponding f of X according to the table is f even odd or neither so pause this video and see if you can figure that out on your own alright now let's work on this together so let's just remind ourselves the definition of even and odd one definition that we can think of is that f of X if f of X is equal to F of negative x then we are dealing with an even function and if f of X is equal to the negative of F of negative x or another way of saying that if F of negative x if F of negative x instead of it being equal to f of X its equal to negative f of X these last two are equivalent that in these situations we are dealing with an odd function and if neither of these are true then we're dealing with neither so what about what's going on over here so let's see f of negative 7 is equal to negative 1 what about F of the negative of negative 7 well that would be f of 7 and we see f of 7 here is also equal to negative 1 so at least in that case in that case if we think of X is 7 f of X is equal to F of negative x so it works for that it also works for negative 3 & 3 f of 3 is equal to F of negative 3 they're both equal to 2 and you can see and you can kind of visualize in your head that we have this symmetry around the y-axis and so this looks like an even function so I will circle that in let's do another example so here once again the table defines function f it's a different function f is this function even odd or neither so pause this video and try to think about it alright so let's just try a few examples so here we have f of 5 is equal to 2 F of 5 is equal to 2 what is f of negative five F of negative 5 not only is it not equal to 2 it would have to be equal to 2 this was an even function and it would be equal to negative two if this was an odd function but it's neither so we very clearly see just looking at that data point that this can neither be even nor odd so I would say neither or neither right over here let's do one more example once again the table defines function f according to the table is it even odd or neither pause the video again try to answer it alright so actually let's just start over here so we have f of 4 is equal to negative 8 what is F of negative 4 and the whole idea here is I want to say okay if f of X is equal to something what is F of negative x well they luckily give us F of negative 4 it is equal to 8 so it looks like it's not equal to f of X its equal to the negative of f of X this is equal to the negative of f of 4 so on that data point alone at least that data point satisfies it being odd it's equal to the negative of f of X but now let's try the other points just to make sure so f of 1 is equal to 5 what is F of negative 1 well it is equal to negative 5 once again F of negative x is equal to the negative of f of X so that checks out and then F of 0 well f of 0 is of course equal to 0 but of course if you say what is the negative of f of if you say what the f of negative of 0 well that's still F of 0 and then if you were to take the negative of 0 that's still 0 so you could view this this is consistent still with being odd this you could view as the negative of F of negative 0 which of course is still going to be 0 so this one is looking pretty good that it is odd