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We should assume that both n and m are integers greater than 1.

CCSS.Math:

let f of X equal a times X to the N plus BX plus 12 over C times X to the M plus DX plus 12 where m and n are integers and ABC and D are unknown constants alright this is interesting which of the following is a process is a possible graph of y is equal to f of X and they tell us the dashed lines indicate asymptotes and they give us four choices here and so I encourage you like always pause this video give a go at it see if you can figure out which of these graphs could be the graph for y equals f of X where f of X where they've given us this information about f of X all right now let's work through this again now this is really interesting it's you know they didn't give us a lot they didn't even tell us what the exponents on X are they haven't even given us the coefficients all they've told all they've told us is this 12 year so this 12 looks like a pretty big clue so what can that tell us the way they've written it we really can't deduce any zeros it for the function we really can't deduce what X values make the numerator equals zero or what X values make the denominator equals zero so it's going to be hard for us to deduce what are the zeros of the function or what are the removable discontinuities or what are the vertical asymptotes but just this 12 sitting here does tell us one thing what happens when x equals zero because when x equals zero every other term in this rational expression is just going to be equal to zero and so we can figure out F of 0 F of 0 is going to be equal to well a times 0 to the N well that's just going to be 0 plus B times 0 well that's just going to be 0 plus 12 over C times 0 to the M power well that's just going to be 0 D times 0 is going to be 0 and then we have our 12 there so we're actually able to figure out what f of 0 is it's 12 over 12 or 1 so we actually know the y intercept for this function let's see if that's enough information for us to figure out if any of these choices could be the graph of I equals f of X so let's see here so choice a our y-intercept is at two when x equals zero our graph goes through two so we can rule that out the y-intercept needs to be 1 so we can rule out let's see choice B does have a y-intercept it looks like just eyeballing it at 1 x equals 0 y is 1 so this looks interesting choice C has its y-intercept at y is equal to negative 1 so once again we can rule that out and choice D has no y-intercept at all so that was enough information and lucky for us because they really didn't give us a lot more information than just about being able to evaluate f of 0 we weren't able to figure out any of the other you know the zeros or the vertical asymptotes or the removable discontinuities so we definitely feel good about that choice