Comparing functions: shared features

which of the features are shared by f of X and G of X select all that apply so they give us f of X as being defined as X to the third minus X and they define G of X essentially with this graph so what are our options so the first one is that they are both odd so right just by looking at G of X you can tell that it is not odd the biggest giveaway is that an odd function would go through the origin G of 0 would have to be equal to 0 if you want to go straight to the definition of an odd function G of X would have to be equal to the negative of G of negative x so for example G of 3 looks like it is for G of 3 is equal to 4 in order for it to be odd the negative or in order for it to be odd G of negative 3 would have to be equal to negative 4 but we see that G of negative 3 is not equal to negative 4 so this one is definitely not odd so this statement can be true they both can't be odd so that's not right they share an x intercept so G of X only has one x-intercept it intersects the x axis right over here at x equals negative 3 now let's think about the x-intercepts of f of X and to do that we just need to factor this expression f of X is equal to X to the third minus X which is the same thing if we factor an X out of x times x squared minus 1 x squared minus 1 is a difference of squares so we can rewrite that as so we'll write our X first this X and then x squared minus 1 is X plus 1 times X minus 1 so when does f of X equal 0 well f of X is equal to 0 when X is equal to 0 when X is equal to 0 that would make this entire expression 0 when X is equal to negative 1 that would make this term and thus the entire expression 0 and when X is equal to positive 1 that would make this last part 0 which would make this entire product 0 so here are the zeros of f of X and none of these coincide with the zeros of G of X so they don't share they don't share an x-intercept they have the and behavior now this is interesting this is saying what's happening as X gets really really really really large or as X gets really really really really really small so we could just think about it right over here as X gets really really really really really large this X to the third is going to is going to grow much faster than this X term right over here so as X grows really really really really large f of X is going to grow really really really large so the the graph and I don't know exactly to see and I could plot a couple of points but the bottom line is f of X is going to approach infinity as X approaches infinity or f of X approaches infinity as as X approaches infinity or as X gets larger and larger and larger and then what happens as X gets smaller and smaller and smaller if we have really small values of X so really negative values of X I should say once again this right over here is going to dominate so f of X is going to become really negative so f of X is going to approach negative infinity as X approaches negative infinity and that is the same behavior of G of X as X approaches a really large value G of X approaches a really large value maybe not as fast as f of X but it still approaches it and likewise as X decreases so does G of X decrease it doesn't decrease as fast as f of X might but it's still going to decrease so they do seem to have the same end behavior at least based on the way that we thought about it just now now the last option is they have a relative maximum at the same x-value so we have to think about what the maximum points are well actually we already know that this is not true because G of X has no relative maximum points in order to have a maximum point you would have to do something like this this right over here would be a relative maximum where you could say a local maximum point it's larger than all of the points around it but eventually the function does surpass it but this right over here has no no local maximum or relative Maxima or little bumps in it G of X doesn't have any so they can't have relative maximum same value so this also is not an option