Even and odd functions: Graphs

which of these functions is odd and so just let's remind ourselves what it means for a function to be odd so I have a function well they've already used F G and H so I'll use J so a function J is odd if you evaluate J at some value so let's say J of a and if you evaluate that J at the negative of that value and if these two things are the negative of each other then my function is odd if these two things were the same if you didn't have this negative here then it would be an even function so let's see which of these meet the criteria of being odd so let's for let's look at f of X so we could pick a particular point so let's say when X is equal to 2 so we get F of 2 f of 2 is equal to 2 now what is F of negative 2 f of negative 2 looks like it is 6 f of negative 2 is equal to 6 so these aren't the negative of each other in order for this to be odd f of negative 2 would have had to be equal to the negative of this would have had to be equal to negative 2 so f of X f of X is definitely not odd so all I do have to do is find even one case with that that violated this constraint to be odd and so I can sits definitely not odd now let's look at G of X so let's look at G of X so I could use it use the same let's see when X is equal to 2 we get G of 2 is equal to negative 7 now let's look at when G is negative 2 so we get G of negative 2 G of negative 2 is also equal to negative 7 so here we have a situation and it looks like that's the case for any X we pick that G of X is going to be equal to G of negative x so G of X is equal to G of negative x its symmetric around the Y or I should say the vertical axis right over here so G of X is even not odd so which of these functions is odd definitely not G of X so our last hope is H of X let's see if H of X seems to meet the criteria so if we do it in this green color so if we take H of 1 H of 1 and we can look at it even visually so H of 1 gets us right over here H of negative 1 seems to get us an equal amount an equal distance negative so it seems to fit for 1 for 2 well 2 is at the at the x-axis but that's definitely H of 2 is 0 H of negative 2 is 0 but those are the negatives of each other 0 is equal to negative 0 if we go to say H of 4 H of 4 is this negative number and H of negative 4 is seems to be a positive number of the same magnitude so once again this is the negative of this so it looks like this is indeed an odd function and another way to visually spot an odd function is a function it's going to go through the origin and you could essentially flip it over on both axes so if you flip this the right half over over the left half and then flip that over the horizontal axis you are going to get this right over here so you see here we're going up into the right here we're going to go down into the left and then you curve right over there you curve up just like that but the easiest way to test it is just to do what we did look at a given X so for example when X is equal to 8 H of 8 looks like this number right around 8 H of negative 8 looks like it's pretty close to negative 8 so they seem to be the negative of each other it looks ounds like a car crash just happened outside anyway hopefully you enjoyed that not the car crash the math problem